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NUMBER 1

THE BELL SYSTEM

TECHNICAL JOURNAL

DEVOTED TO THE SCIENTIHC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION

Stabilized Feedback Amplifiers — H. S. Black ... 1

Open-Wire Crosstalk — A. G. Chapman 19

Vacuum Tube Electronics at Ultra-high Frequencies —

F. B. Llewellyn 59

Contemporary Advances in Physics, XXVII — The

Nucleus, Second Part — Karl K. Darrow .... 102

Abstracts of Technical Papers 159

Contributors to this Issue 161

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

NEW YORK

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THE BELL SYSTEM TECHNICAL JOURNAL

Published quarterly by the

American Telephone and Telegraph Company

195 Broadway ^ New York, N. F.

miiiiiuiiHiiniiiiiiiiiiiiiiiiiiiili

Bancroft Gherardi
L. F. Morehouse
D. Levinger

EDITORIAL BOARD

H. P. Charlesworth
E. H. Colpitts
O. E. Buckley

F. B. Jewett

O. B. BlackweU

H. S. Osborne

Philander Norton, Editor

J. O. Perrine, Associate Editor

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PRINTED IN O. 8. A.

THE BELL SYSTEM

TECHNICAL JOURNAL

A JOURNAL DEVOTED TO THE

SCIENTIFIC AND ENGINEERING

ASPECTS OF ELECTRICAL

COMMUNICATION

EDITORIAL BOARD

Bancroft Gherardi H. P. Charlesworth F. B. Jewett

L. F. Morehouse E. H. Colpitts O. B. Blackwell

D. Levinger O. E. Buckley H. S. Osborne

Philander Norton, Editor J. O. Perrine, Associate Editor

TABLE OF CONTENTS

AND

INDEX

VOLUME XIII
1934

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

NEW YORK

Periodical

PRINTED IN U. S. A.

THE BELL SYSTEM

TECHNICAL JOURNAL

VOLUME XIII, 1934
Table of Contents

January, 1934

Stabilized Feedback Amplifiers — H. S. Black 1

Open-Wire Crosstalk — A. G. Chapman 19

Vacuum Tube Electronics at Ultra-high Frequencies —

F. B. Llewellyn 59
Contemporary Advances in Physics, XXVII — The Nucleus,

Second Part — Karl K. Darrow 102

April, 1934

The Carbon Microphone: An Account of Some Researches Bear-
ing on Its Action — F. S. Goticher 163

Open -Wire Crosstalk — A. G. Chapman 195

Symposium on Wire Transmission of Symphonic Music and Its
Reproduction in Auditory Perspective:

Basic Requirements — Harvey Fletcher 239

Physical Factors — /. C. Steinberg and W. B. Snow 245

Loud Speakers and Microphones — E. C. Wenteand A. L. Thuras 259

Amplifiers — E. 0. Scriven 278

Transmission Lines — H. A. Affel, R. W. Chesnut and R. 11. Mills 285
System Adaptation — E. II. Bedell and Iden Kerney 301

BELL SYSTEM TECHNICAL JOURNAL

JULY, 1934

The Compandor — An Aid Against Static in Radio Telephony —

R. C. Mathes and S. B. Wright 315

The Effect of Background Noise in Shared Channel Broad-
casting— C. B. Aiken 3?)?)

Wide-Band Open-Wire Program System — //. 5. Hamilton 351

Line Filter for Program System — A. W. Clement 382

Contemporary Advances in Physics, XXVIII — The Nucleus,

Third Part— Xar/ K. Darrow 391

Electrical Wave Filters Employing Quartz Crystals as Ele-
ments— W. P. Mason 405

Some Improvements in Quartz Crystal Circuit Elements —

F. R. Lack, G. W. Willard and I. E. Fair 453

A Theory of Scanning and Its Relation to the Characteristics of
the Transmitted Signal in Telephotography and Television —

Pierre Mertz and Frank Gray 464

October, 1934

An Extension of the Theory of Three-Electrode Vacuum Tube

Circuits — S. A. Levin and Liss C. Peterson 523

The Electromagnetic Theory of Coaxial Transmission Lines and

Cylindrical Shields — S. A. Schelktinoff 532

Contemporary Advances in Physics, XXVIII — The Nucleus,

Third Part — Karl K. Darrow 580

The Measurement and Reduction of Microphonic Noise in Vac-
uum Tubes — D. B. Penick 614

Fluctuation Noise in Vacuum Tubes — G. L. Pearson 634

Systems for Wide-Band Transmission Over Coaxial Lines —

L. Espenschied and M. E. Striehy 654

Regeneration Theory and Experiment —

E. Peterson, J. G. Kreer and L. A. Ware 680

Index to Volume XIII

Affel, H. A., R. W. Chesnut and R. H. Mills, Transmission Lines, page 285.

Aiken, C. B., The Effect of Background Noise in Shared Channel Broadcasting,

page 333.
Amplifiers, E. O. Scriven, page 278.
Amplifiers, Stabilized Feedback, H. S. Black, page 1.
Auditory Perspective, Symposium on Wire Transmission of Symphonic Music and

Its Reproduction in, pages 239-301.

Basic Requirements (of Wire Transmission of Symphonic Music and Its Reproduction
in Auditory Perspective), Harvey Fletcher, page 239.

Bedell, E. H. and Iden Kerney, System Adaptation (of Symposium on Wire Trans-
mission of Symphonic Music and Its Reproduction in Auditory Perspective),
page 301.

Black, H. S., Stabilized Feedback Amplifiers, page 1.

Broadcasting, Shared Channel, The Effect of Background Noise in, C. B. Aiken, page
333.

Chapmayi, A. C, Open-Wire Crosstalk, pages 19 and 195.

Chesnut, R. W., H. A. Affel and R. H. Mills, Transmission Lines, page 285.

Clemettt, A. W., Line Filter for Program System, page 382.

Coaxial Lines, Systems for Wide-Band Transmission Over, L. Espenschied and M. E.

Strieby, page 654.
Coaxial Transmission Lines and Cylindrical Shields, The Electromagnetic Theory of,

5. A. Schelkunoff, page 532.
Compandor, The — An Aid Against Static in Radio Telephony, R. C. Mathes and

S. B. Wright, page 315.
Contemporary Advances in Physics, XXVII — The Nucleus, Second Part, Karl K.

Darrow, page 102.
Contemporary Advances in Physics, XXVIII — The Nucleus, Third Part, Karl K.

Darrow, pages 391 and 580.
Crosstalk, Open- Wire, A. G. Chapman, pages 19 and 195.

Darrow, Karl K., Contemporary Advances in Physics, XXVII — The Nucleus,
Second Part, page 102.
Contemporary Advances in Physics, XXVIII — The Nucleus, Third Part, pages
391 and 580.

Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields, The,

5. A. Schelkunoff, page 532.
Espenschied, L. and M. E. Strieby, Systems for Wide-Band Transmission Over

Coaxial Lines, page 654.

Fair, I. E., F. R. Lack and G. W. Willard, Some Improvements in Quartz Crystal

Circuit Elements, page 453.
Filter, Line, for Program System, A. W. Clement, page 382.

BELL SYSTEM TECHNICAL JOURNAL

Filters, Electrical Wave, Employing Quartz Crystals as Elements, W. P. Mason

page 405.
Fletcher, Harvey, Basic Requirements (of Wire Transmission of Symphonic Music

and its Reproduction in Auditory Perspective), page 239.

Gaucher, F. S., The Carbon Microphone: An Account of Some Researches Bearing
on its Action, page 163.

Gray, Frank and Pierre Mertz, A Theory of Scanning and Its Relation to the Char-
acteristics of the Transmitted Signal in Telephotography and Television,
page 464.

Hamilton, H. S., Wide-Band Open-Wire Program System, page 351.

Kerney, Men and E. H. Bedell, System Adaptation (of Symposium on Wire Trans-
mission of Symphonic Music and Its Reproduction in Auditory Perspective),
page 301.

Kreer, J. G., L. A. Ware and E. Peterson, Regeneration Theory and Experiment,
page 680.

Lack, F. R., G. W. Willard and I. E. Fair, Some Improvements in Quartz Crystal

Circuit Elements, page 453.
Levin, S. A. and Liss C. Peterson, An Extension of the Theory of Three-Electrode

Vacuum Tube Circuits, page 523.
Llewellyn, F. B., Vacuum Tube Electronics at Ultra-high Frequencies, page 59.
Loud Speakers and Microphones, E. C. Wejite and A. L. Thuras, page 259.

Mason, W. P., Electrical Wave Filters Employing Quartz Crystals as Elements,

page 405.
Mathes, R. C. and S. B. Wright, The Compandor — An Aid Against Static in Radio

Telephony, page 315.
Mertz, Pierre and Frank Gray, A Theory of Scanning and Its Relation to the Char-
acteristics of the Transmitted Signal in Telephotography and Television,

page 464.
Microphone, The Carbon: An Account of Some Researches Bearing on Its Action.

F. S. Gaucher, page 163.
Microphones, Loud Speakers and, E. C. Wente and A. L. Thuras, page 259.
Microphonic Noise in Vacuum Tubes, The Measurement and Reduction of, D. B.

Penick, page 614.
Mills, R. H., H. A. Affel and R. W. Chesnut, Transmission Lines, page 285.
Music, Symphonic, Symposium on Wire Transmission of and Its Reproduction in

Auditory Perspective, pages 239-301.

Noise, Background, The Effect of in Shared Channel Broadcasting, C. B. Aiken,

page 333.
Noise in Vacuum Tubes, Fluctuation, G. L. Pearson, page 634.

Noise, Microphonic, The Measurement and Reduction of in Vacuum Tubes, D. B.
Penick, page 614.

Pearson, G. L., Fluctuation Noise in Vacuum Tubes, page 634.

Penick, D. B., The Measurement and Reduction of Microphonic Noise in X'acuum
Tubes, page 614.

BELL SYSTEM TECHNICAL JOURNAL

Peterson, E., J. G. Kreer and L. A. Ware, Regeneration Theory and Experiment,

page 680.
Peterson, Liss C. and S. A. Levin, An Extension of the Theory of Three-Electrode

Vacuum Tube Circuits, page 523.
Physical Factors (of Wire Transmission of Symphonic Music and Its Reproduction

in Auditory Perspective), /. C. Steinberg and W. B. Snow, page 245.
Physics, XXVII, Contemporary Advances in — The Nucleus, Second Part, Karl K.

Darrow, page 102.
Physics, XXVIII, Contemporary Advances in — The Nucleus, Third Part, Karl K.

Darrow, pages 391 and 580,

Quartz Crystal Circuit Elements, Some Improvements in, F. R. Lack, G. W. Willard

and I. E. Fair, page 453.
Quartz Crystals in Elejnents, Electrical Wave Filters Employing, W. P. Mason,

page 405.

Radio: Line Filter for Program System, A. W. Clement, page 382,

Radio: The Efifect of Background Noise in Shared Channel Broadcasting, C. B.

Aiken, page iii.
Radio: Wide-Band Open- Wire Program System, H. S. Hamilton, page 351.
Radio Telephony, The Compandor — An Aid Against Static in, R. C. Mathes and S. B.

Wright, page 315.
Regeneration Theory and Experiment, E. Peterson, J. G. Kreer, and L. A. Ware,

page 680.

Schelkunoff, S. A., The Electromagnetic Theory of Coaxial Transmission Lines and
Cylindrical Shields, page 532.

Scriven, E. O., Amplifiers, page 278.

Snow, W. B. and J. C. Steinberg, Physical Factors (of Wire Transmission of Sym-
phonic Music and Its Reproduction in Auditory Perspective), page 245.

Steinberg, J. C. and W. B. Snow, Physical Factors (of Wire Transmission of Sym-
phonic Music and Its Reproduction in Auditory Perspective), page 245.

Strieby, M. E. and L. Espenschied, Systems for Wide-Band Transmission Over Coaxial
Lines, page 654,

Tele,photography and Television, A Theory of Scanning and Its Relation to the

Characteristics of the Transmitted Signal in, Pierre Mertz and Frank Gray,

page 464.
Television, A Theory of Scanning and Its Relation to the Characteristics of the

Transmitted Signal in Telephotography and, Pierre Mertz and Frank Gray,

page 464.
Thuras, A. L. and E. C. Wente, Loud Speakers and Microphones, page 259.
Transmission Lines, H. A. Affel, R. W. Chesnut and R. H. Mills, page 285.

Vacuum Tube Circuits, Three-Electrode, An Extension of the Theory of, S. A. Levin

and Liss C. Peterson, page 523.
Vacuum Tube Electronics at Ultra-high Frequencies, F. B. Llewellyn, page 59.
Vacuum Tubes, Fluctuation Noise in, G. L. Pearson, page 634.
Vacuum Tubes, The Measurement and Reduction of Microphonic Noise in, D. B.

Penick, page 614.

BELL SYSTEM TECHNICAL JOURNAL

Ware, L. A., E. Peterson and J. G. Kreer, Regeneration Theory and Experiment, page
680.

Wente, E. C. and A. L. Thuras, Loud Speakers and Microphones, page 259.

Wide-Band Open-Wire Program System, H. S. Hamilton, page 351.

Wide-Band Transmission Over Coaxial Lines, Systems for, L. Espenschied and M. E.
Strieby, page 654.

Wide Band: The Electromagnetic Theory of Coaxial Transmission Lines and Cy-
lindrical Shields, S. A. Schelkunoff, page 532.

Wide Band: Symposium on Wire Transmission of Symphonic Music and Its Repro-
duction in Auditory Perspective, pages 239-301.

Wide Band: Stabilized Feedback Amplifiers, H. S. Black, page 1.

Willard, G. W., I. E. Fair and F. R. Lack, Some Improvements in Quartz Crystal
Circuit Elements, page 453.

Wright, S. B. and R. C. Mathes, The Compandor— An Aid Against Static in Radio
Telephony, page 315.

The Bell System Technical Journal

January, 1934

Stabilized Feedback Amplifiers*

By H. S. BLACK

This paper describes and explains the theory of the feedback principle
and then demonstrates how stability of amplification and reduction of
modulation products, as well as certain other advantages, follow when
stabilized feedback is applied to an amplifier. The underlying principle
of design by means of which singing is avoided is next set forth. The paper
concludes with some examples of results obtained on amplifiers which have
been built employing this new principle.

The carrier-in-cable system dealt with in a companion paper ^ involves
many amplifiers in tandem with many telephone channels passing through
each amplifier and constitutes, therefore, an ideal field for application of
this feedback principle. A field trial of this system was made at Morris-
town, New Jersey, in which seventy of these amplifiers were operated in
tandem. The results of this trial were highly satisfactory and demon-
strated conclusively the correctness of the theor>' and the practicability
of its commercial application.

Introduction

DUE TO advances in vacuum tube development and amplifier
technique, it is now possible to secure any desired amplification
of the electrical waves used in the communication field. When many
amplifiers are worked in tandem, however, it becomes difficult to keep
the overall circuit efficiency constant, variations in battery potentials
and currents, small when considered individually, adding up to produce
serious transmission changes for the overall circuit. Furthermore,
although it has remarkably linear properties, when the modern vacuum
tube amplifier is used to handle a number of carrier telephone channels,
extraneous frequencies are generated which cause interference between
the channels. To keep this interference within proper bounds involves
serious sacrifice of effective amplifier capacity or the use of a push-pull
arrangement which, while giving some increase in capacity, adds to
maintenance difficulty.

However, by building an amplifier whose gain is deliberately made,
say 40 decibels higher than necessary (10,000 fold excess on energy
basis), and then feeding the output back on the input in such a way

* Presented at Winter Convention of A. I. E. E., New York City, Jan. 23-26,
1934. Published in Electrical Engineering, January, 1934.

' "Carrier in Cable" by A. B. Clark and B. W. Kendall, presented at the A. I. E. E.
Summer Convention, Chicago, 111., June, 1933; published in Electrical Engineering,
July. 1933, and in Bell Sys. Tech. Jour., July, 1933.

2 BELL SYSTEM TECHNICAL JOURNAL

as to throw away the excess gain, it has been found possible to effect
extraordinary improvement in constancy of amplification and freedom
from non-linearity. By employing this feedback principle, amplifiers
have been built and used whose gain varied less than 0.01 db with a
change in plate voltage from 240 to 260 volts and whose modulation
products were 75 db below the signal output at full load. For an
amplifier of convc ntional design and comparable size this change in
plate voltage would have produced about 0.7 db variation while the
modulation products would have been only 35 db down; in other
words, 40 db reduction in modulation products was effected. (On an
energy basis the reduction was 10,000 fold.)

Stabilized feedback possesses other advantages including reduced
delay and delay distortion, reduced noise disturbance from the power
supply circuits and various other features best appreciated by practical
designers of amplifiers.

It is far from a simple proposition to employ feedback in this way
because of the very special control required of phase shifts in the
amplifier and feedback circuits, not only throughout the useful fre-
quency band but also for a wide range of frequencies above and below
this band. Unless these relations are maintained, singing will occur,
usually at frequencies outside the useful range. Once having achieved
a design, however, in which proper phase relations are secured, expe-
rience has demonstrated that the performance obtained is perfectly
reliable.

Circuit Arrangement

In the amplifier of Fig. 1, a portion of the output is returned to the
input to produce feedback action. The upper branch, called the
/x-circuit, is represented as containing active elements such as an
amplifier while the lower branch, called the j8-circuit, is shown as a
passive network. The way a voltage is modified after once traversing
each circuit is denoted /x and ^ respectively and the product, ^i/3, repre-
sents how a voltage is modified after making a single journey around
amplifier and feedback circuits. Both /x and j8 are complex quantities,
functions of frequency, and in the generalized concept either or both
may be greater or less in absolute value than unity.^

Figure 2 shows an arrangement convenient for some purposes where,
by using balanced bridges in input and output circuits, interaction
between input and output is avoided and feedback action and amplifier
impedances are made independent of the properties of circuits con-
nected to the amplifier.

* /x is not used in the sense that it is sometimes used, namely, to denote the
amplification constant of a particular tube, but as the complex ratio of the output
to the input voltage of the amplifier circuit.

STABILIZED FEEDBACK AMPLIFIERS

♦■E + N +D

Fig. 1 — Amplifier system with feedback.

e — Signal input voltage.

y. — Propagation of amplifier circuit.

p.e — Signal output voltage without feedback.

n — Noise output voltage without feedback.

d{E) — Distortion output voltage without feedback.

/3 — Propagation of feedback circuit.

E — Signal output voltage with feedback.

N — Noise output voltage with feedback.

D — Distortion output voltage with feedback.

The output voltage with feedback is E -\- N -\- D and is the sum of fxe -\- n -\- d{E),
the value without feedback plus yu/3[£ + N + D] due to feedback.

E + N + D=iJie + 7i + d{E) + n^lE + N + D^

IE + N + Z)](l - M/3) = fxe + n + d{E)

E + N + D

fie

d{E)

1 - M/3 1 - M^ 1 - M/8

If |ju/3| ^ 1, £ = — -. Under this condition the amplification is independent of

IX but does depend upon /3. Consequently the over-all characteristic will be con-
trolled by the feedback circuit which may include equalizers or other corrective
networks.

General Equation

In Fig. 1, jS is zero without feedback and a signal voltage, Bq, applied
to the input of the /x-circuit produces an output voltage. This is
made up of what is wanted, the amplified signal, Eq, and components
that are not wanted, namely, noise and distortion designated Nq and
Dq and assumed to be generated within the amplifier. It is further
assumed that the noise is independent of the signal and the distortion
generator or modulation a function only of the signal output. Using the
notation of Fig. 1 , the output without feedback may be written as :

Eq + Nq + Do = ixeo + n + d(Eo), (1)

where zero subscripts refer to conditions without feedback.

BELL SYSTEM TECHNICAL JOURNAL

STABILIZED FEEDBACK AMPLIFIERS 5

With feedback, fi is not zero and the input to the ^-circuit becomes
eo + i8(£ + N +'D). The output is E + A^ + /) and is equal to
M[go + KE + N + D)'] + n + d{E) or:

In the output, signal, noise and modulation are divided by (1 — miS)i
and assuming 1 1 — ;U/3 1 > 1, all are reduced.

Change in Gain Due to Feedback

From equation (2), the amplification with feedback equals the
amplification without feedback divided by (1 — ai/S). The effect of
adding feedback, therefore, usually is to change the gain of the amplifier
and this change will be expressed as:

GcF = 20 logi

1 -M/3

(3)

where Gcf is dh change in gain due to feedback. 1/(1 — ix^) will be used
as a quantitative measure of the effect of feedback and the feedback
referred to as positive feedback or negative feedback according as the
absolute value of 1/(1 — m/3) is greater or less than unity. Positive
feedback increases the gain of the amplifier; negative feedback reduces
it. The term feedback is not limited merely to those cases where the
absolute value of 1/(1 — ii0) is other than unity.
From /ijS = | mi8 I [$ and (3) , it may be shown that :

10-OcF/io = 1 - 2!m/3| COS* + |m/3|-, (4)

which is the equation for a family of concentric circles of radii
10~^cf/io about the point 1, 0. Figure 3 is a polar diagram of the
vector field of m/? = Im/SI |$. Using rectangular instead of polar
coordinates, Fig. 4 corresponds to Fig. 3 and may be regarded as a
diagram of the field of /x/3 where the parameter is db change in gain
due to feedback. From these diagrams all of the essential properties
of feedback action can be obtained such as change in amplification,
effect on linearity, change in stability due to variations in various
parts of the system, reduction of noise, etc. Certain significant
boundaries have been designated similarly on both figures.

For example, boundary A is the locus of zero change in gain due to
feedback. Along this parametric contour line where the absolute
magnitude of amplification is not changed by feedback action, values
of I )u/3 1 range from zero to 2 and the phase shift, $, around the amplifier

6 BELL SYSTEM TECHNICAL JOURNAL

and feedback circuits equals cos~^ |A'i3|/2 and, therefore, lies between
— 90° and + 90°. For all conditions inside or above this boundary,
the gain with feedback is increased; outside or below, the gain is
decreased.

Stability

From equation (2), mV(1 ~ Mi3) is the amplified signal with feedback
and, therefore, ^/(l — m/3) is an index of the amplification. It is of
course a complex ratio. It will be designated Ap and referred to as
the amplification with feedback.

To consider the effect of feedback upon stability of amplification,
the stability will be viewed as the ratio of a change, hAp, to Af where
hAp is due to a change in either a* or j3 and the effects may be derived
by assuming the variations are small.

Ap =

1 -;z/3'

bu.

' bAp'

L Af \

. 1

-/./?'

4/
ip

AC/:
1 -

(5)

(6)
(7)

If /i/3 :^ 1, it is seen that p, or the ^t-circuit is stabilized by an amount
corresponding to the reduction in amplification and the effect of intro-
ducing a gain or loss in the /^-circuit is to produce no material change
in the overall amplification of the system ; the stability of amplification
as affected by j8 or the jS-circuit is neither appreciably improved nor
degraded since increasing the loss in the /S-circuit raises the gain of
the amplifier by an amount almost corresponding to the loss intro-
duced and vice-versa. If /x and /3 are both varied and the variations
sufficiently small, the effect is the same as if each were changed sepa-
rately and the two results then combined.

In certain practical applications of amplifiers it is the change in
gain or ammeter or voltmeter reading at the output that is a measure
of the stability rather than the complex ratio previously treated. The
conditions surrounding gain stability may be examined by considering
the absolute value of ^f- This is shown as follows: Let {dh) represent
the gain in decibels corresponding to A p. Then

{dh) = 20 1ogio \Ap\,
b\Ap\

8(db) = 8.686

\A.

(8)

STABILIZED FEEDBACK AMPLIFIERS
To get the absolute value of the amplification: Let

ixfi = l/x/31 I*,

\Af\ I/^I

VI - 2|mi3| cos* + |/xj8|2

The stability of amplification which is proportional to the
stability is given by:

\Af\
b\AF\

\Af\
5Uf|

1 -

- Im^I

cos <i>

plMll

iMi |l-;"/3|^

MiS

cos $ — ;u/3

p;/^!]

1^1

1 -M/3

L |1-M/3|

L i/^i J

1 -A

_ |]

sin $

[5*].

(9)
(10)

gain

(11)
(12)
(13)

Fig. 3 — The vector field of ///i. See caption for Fig. 4.

BELL SYSTEM TECHNICAL JOURNAL

30°

<0 120
<

y 140

LU o

Q 150

i^H 160°
170°
180°

]

[\IM/

///

— «/-
o/

i\\

^A^x^<^

)^..

'/f

-^a3

>r:

>i^

^^rJf:

rx'

= cos - ( ^■p'

•/

ii'i

cot

340°

330°

320°

310°

1-
<

300°

I.)

rtl

290°

280°

270°

rr.

260

L-

250°

</)

240°

UJ
<0

230

220

210°

UJ
Q

200°

^eH

190°

.8 1.0

1.4 1.6

1.8 2.0 2.2 2.4 2.6 ^a 3.0

Fig. 4 — Phase shift around the feedback path plotted as a function of |m/3| ,
the absolute value of n0.

IJ.I3 is a complex quantity which represents the ratio by which the amplifier and

feedback (or more generally ix and /3) modify a voltage in a single trip around the

closed path.

First, there is a set of boundary curves indicated as A, B, C, D, E, F, G, H, I, and /

which gives either limiting or significent values of |ju/3| and <i>.

Secondly, there is a family of curves in which db change in gain due to feedback is

the parameter.

Boundaries

A. Conditions in which gain and modulation are unaffected by feedback.

B. Con3tant amplification ratio against small variations in |/3].

Constant change in gain, t- r-r , against variations in \fx\ and |/3|.

I 1 — MP I
Stable phase shift through the amplifier against variations in ^g-
The boundary on which the stability of amplification is unaffected by feedback.

C. Constant amplification ratio against small variations in |/x|.
Constant phase shift through the amplifier against variations in ^/x.

The absolute magnitude of the voltage ted bark tt-^ — jr is constant against

\-ariations in |/.i] and \/3\ .

STABILIZED FEEDBACK AMPLIFIERS 9

A curious fact to be noted from (11) is that it is possible to choose
a value of m/S (namely, |ju|3| = sec $) so that the numerator of the
right hand side vanishes. This means that the gain stability is
perfect, assuming differential variations in \fx\. Referring to Figs. 3
and 4, contour C is the locus of |a£/3| = sec <l> and it includes all ampli-
fiers whose gain is unaffected by small variations in | ^i | . In this way
it is even possible to stabilize an amplifier whose feedback is positive,
i.e., feedback may be utilized to raise the gain of an amplifier and, at
the same time, the gain stability with feedback need not be degraded
but on the contrary improved. If a similar procedure is followed
with an amplifier whose feedback is negative, the gain stability will
be theoretically perfect and independent of the reductions in gain due
to feedback. Over too wide a frequency band practical difficulties
will limit the improvements possible by these methods.

With negative feedback, gain stability is always improved by an
amount at least as great as corresponds to the reduction in gain and
generally more; with positive feedback, gain stability is never degraded
by more than would correspond to the increase in gain and under
appropriate conditions, assuming the variations are not too great,
is as good as or much better than without feedback. With positive
feedback, the variations in /i or /3 must not be permitted to become
sufficiently great to cause the amplifier to sing or give rise to instabil-
ity as defined in a following section on "Avoiding Singing."

Modulation

To determine the effect of feedback action upon modulation pro-
duced in the amplifier circuit, it is convenient to assume that the
output of undistorted signal is made the same with and without feed-
back and that a comparison is then made of the difference in modula-
tion with and without feedback. Therefore, with feedback, the input
is changed to e = go(l — m/3) and, referring to equation (2), the out-
put voltage is m^o. and the generated modulation, d(E), assumes its
value without feedback, d(Eo),and d(E)l{l -/i/3) becomes d(Eo) / (1 - (il3)
which is Dol{l — jj.^). This relationship is approximate because the

D. |m;S| = 1.

£. * = 90°. Improvement in gain stability corresponds to twice db reduction

in gain.
F and G. Constant amplification ratio against variations in <i>.

Constant phase shift through the amplifier against variations in |/x| and
l/3i.
H. Same properties as B.
I. Same properties as E.

J. Conditions in which -r- L— -r = -r— T the overall gain is the exact negative

I 1 — MP I I P I

in^■erse of the transmission through the /3-circuit.

10 BELL SYSTEM TECHNICAL JOURNAL

voltage at the input without feedback is free from distortion and with
feedback it is not and, hence, the assumption that the generated
modulation is a function only of the signal output used in deriving
equation (2) is not necessarily justified.

From the relationship D = -Do/(l — m/^), it is to be concluded that
modulation with feedback will be reduced db for db as the effect of
feedback action causes an arbitrary db reduction in the gain of the
amplifier, i.e., when the feedback is negative. With positive feedback
the opposite is true, the modulation being increased by an amount
corresponding to the increase in amplification.

If modulation in the j3-circuit is a factor, it can be shown that
usually in its effect on the output, the modulation level at the output
due to non-linearity of the /3-circuit is approximately /x/5/(l — At/S) multi-
plied by the modulation generated in the /3-circuit acting alone and

without feedback.

Additional Effects

Noise

A criterion of the worth of a reduction in noise is the reduction in
signal-to-noise ratio at the output of an amplifier. Assuming that
the amount of noise introduced is the same in two systems, for example
with and without feedback respectively, and that the signal outputs
are the same, a comparison of the signal-to-noise ratios will be affected
by the amplification between the place at which the noise enters and
the output. Denoting this amplification by a and ao respectively, it
can be shown that the relation between the two noise ratios is
{ao/a){l — MiS). This is called the noise index.

If noise is introduced in the power supply circuits of the last tube,
ao/a = 1 and the noise index is (1 — m/3)- As a result of this relation
less expensive power supply filters are possible in the last stage.

Phase Shift, Envelope Delay, Delay Distortion
In the expression Af = [m/(1 — m/3)] [£. ^ is the overall phase shift
with feedback, and it can be shown that the phase shift through the
amplifier with feedback may be made to approach the phase shift through
the ^-circuit plus 180 degrees. The effect of phase shift in the jS-circuit
is not correspondingly reduced. It will be recalled that in reducing
the change in phase shift with frequency, envelope delay, which
is the slope of the phase shift with respect to the angular velocity,
(J, = 27rf, also is reduced. The delay distortion likewise is reduced
because a measure of delay distortion at a particular frequency is the
difference between the envelope delay at that frequency and the least
envelope delay in the band.

STABILIZED FEEDBACK AMPLIFIERS

^-Circuit Equalization
Referring to equation (2), the output voltage, E, approaches — eo/iS

as 1 — yu/3 = — ;uj8 and equals it in absolute value if cos $ =

2|/x^|

where n^ = 1^/31 [_^- Under these circumstances increasing the loss
in the jS-circuit one db raises the gain of the amplifier one db and vice-
versa, thus giving any gain-frequency characteristic for which a like
loss-frequency characteristic can be inserted in the ^-circuit. This
procedure has been termed /3-circuit equalization. It possesses other
advantages which cannot be dwelt upon here.

Avoid Singing

Having considered the theory up to this point, experimental evidence
was readily acquired to demonstrate that /x/3 might assume large values.

UNSTABLE

Fig. 5 — -Measured m/S characteristics of two amplifiers.

for example 10 or 10,000, provided $ was not at the same time zero.
However, one noticeable feature about the field of /i/3 (Figs. 3 and 4) is
that it implies that even though the phase shift is zero and the absolute
value of /x/3 exceeds unity, self-oscillations or singing will not result.
This may or may not be true. When the author first thought about
this matter he suspected that owing to practical non-linearity, singing
would result whenever the gain around the closed loop equalled or
exceeded the loss and simultaneously the phase shift was zero, i.e.,
m/3 = |ju/3| -f JO ^ 1. Results of experiments, however, seemed to
indicate something more was involved and these matters were de-
scribed to Mr. H. Nyquist, who developed a more general criterion

BELL SYSTEM TECHNICAL JOURNAL

for freedom from instability •' applicable to an amplifier having linear
positive constants.

To use this criterion, plot /x/3 (the modulus and argument vary with
frequency) and its complex conjugate in polar coordinates for all
values of frequency from 0 to + <» . If the resulting loop or loops
do not enclose the point (1,0) the system will be stable, otherwise
not.^ The envelope of the transient response of a stable amplifier

O FEED

BAC

j >

1 c

RA
4

TING R/
-40 KG

^NGt

'—*\

EEDBAC

K 1

-N

„.

. —

—

■EEDBAC
.1 t_

K 2

-— 1—

_,.

— ^—

>.,

■.\\

20-^^

1,000 IQOOO

FREQUENCY- CYCLES

100,000

Fig. 6 — Gain frequency characteristics with and without feedback of amplifier of

Fig. 2.

always dies away exponentially with time; that of an unstable amplifier
in all physically realizable cases increases with time. Characteristics
A and B in Fig. 5 are results of measurements on two different
amplifiers; the amplifier having jujS-characteristic denoted A was stable;
the other unstable.

The number of stages of amplification that can be used in a single
amplifier is not significant except insofar as it affects the question of
avoiding singing. Amplifiers with considerable negative feedback

' For a complete description of the criterion for stability and instability and
exactly what is meant by enclosing the point (1, 0), reference should be made to
"Regeneration Theory" — H. Nyquist, Bell System Technical Journal, Vol. XI,
pp. 126-147, July, 1932.

STABILIZED FEEDBACK AMPLIFIERS

have been tested where the number of stages ranged from one to five
inclusive. In every case the feedback path was from the output of
the last tube to the input of the first tube.

Sx^

^%^

2?y

WITH FEEDE

^^^-"^ 3F

\ l

CRin

POSITIVE

60
50

\ \

/\ '

\ '

N^F

^it

\ \

FEEDBACK

»
1

V.'

HARMONIC

MEASURED A"

- 15

\
\

N
N

10 20 30 40

OUTPUT OF FUNDAMENTAL- MILLIAMPERE:S INTO 600 OHMS

Fig. 7 — Modulation characteristics with and without feedback for the amplifier of

Fig. 2.

Experimental Results

Figures 6 and 7 show how the gain-frequency and modulation char-
acteristics of the three-stage impedance coupled amplifier of Fig. 2
are improved by negative feedback. In Fig. 7, the improvement in
harmonics is not exactly equal to the db reduction in gain. Figure 8

BELL SYSTEM TECHNICAL JOURNAL

shows measurements on a different amplifier in which harmonics are
reduced as negative feedback is increased, db for db over a 65 db range.
That the gain with frequency is practically independent of small vari-
ations in I /x I is shown by Fig. 9. This is a characteristic of the Morris-
town amplifier described in the paper by Messrs. Clark and Kendall ^
which meets the severe requirements imposed upon a repeater amplifier
for use in cable carrier systems. Designed to amplify frequencies from 4 kc

95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20

FUNDAMENTAL OUTPUT HELD CONSTANT
AT 20 MILLIAMPERES INTO 600'^

•

<<>.

rA
/

/
f

•

70 <

111

-I
-)

•2.

5 10 15 20 25 30 35 40 45 50 55 60 65
DB REDUCTION IN GAIN DUE TO FEEDBACK

Fig. 8 — Improvement of harmonics with feedback. One example of another
amplifier in which with 60 db feedback, harmonic currents in the output are only
one-thousandth and their energy one-millionth of the values without feedback.

to 40 kc the maximum change in gain due to variations in plate voltage
does not exceed 7/10000 db per volt and at 20 kc the change is only
1/20000 db per volt. This illustrates that for small changes in |/t|,
the ratio of the stability without feedback to the stability with feed-
back, called the stability index, approaches 1 1 — /i|3| V(l ~ |i"|S| cos $)
and gain stability is improved at least as much as the gain is reduced
and usually more and is theoreticaljy perfect if cos <l> = 1/|m/3|.

^Loc. cit.

STABILIZED FEEDBACK AMPLIFIERS

0.1

I 5 10

FREQUENCY IN KILOCYCLES

?^
Q
LJ UJ
Ul UJ
Zli.

DB CHANGE IN GAIN WITHOUT FEEDBACK
-.2 0 +.2

+ .005

-j005

5£^c

NORMAL OPERATING VOLTAGE
250 ± 2 VOLTS

"''~-~-

"'"■---^^

-20 KC - i

^^^~^-^.,

■^

50.010

50.005 z
<

a.
50.000 y

49.995

24-0 245 250 255 260

PLATE BATTERY SUPPLY VOLTAGE

Fig. 9 — Representative gain stability of a single amplifier as determined by
measuring 69 feedback amplifiers in tandem at Morristown, N. J.

The upper figure shows the absolute value of the stability index. It can be seen
that between 20 and 25 kc the improvement in stability is more than 1000 to 1 yet
the reduction in gain was less than 35 db.

The lower figure shows change in gain of the feedback amplifier with changes in
the plate battery voltage and the corresponding changes in gain without feedback.
At some frequencies the change in gain is of the same sign as without feedback and
at others it is of opposite sign and it can be seen that near 23 kc the stability must
be perfect.

BELL SYSTEM TECHNICAL JOURNAL

Figure 10 indicates the effectiveness with which the gain of a feed-
back ampHfier can be made independent of variations in input ampU-
tude up to practically the overload point of the amplifier. These
measurements were made on a three-stage amplifier designed to work
from 2).2> kc to 50 kc.

Figure 11 shows that negative feedback may be used to improve
phase shift and reduce delay and delay distortion. These measurements

WITHOL

T FEEt

WITH

FEEDBA

8 10 12 14- 16

MILLIAMPERES INTO 600"^

Fig. 10 — Ckiin-load characteristic with and without feedback for a low level aiii])lifier
designed to amplify frequencies from 3.5 to 50 kc.

STABILIZED FEEDBACK AMPLIFIERS

260

24-0

220

200

180

160

I 140

>o ^
o

-o -*

Qd 1

Vl^aS-v TO 6500 'v A

\ ! 1

00
D

\ ' 1

Y-WITHOUr FEEDBACK

.-WITHOU

■ FEE

3BA(

WITH
FEEDBAC

0 100 1000 100

:quency in cycles per secon

T'--

ElEDBAC

(

•n

■>

'■"

20 100 1000 10000 20000

FREQUENCY IN CYCLES PER SECOND

Fig. 11 — Phase shift, delay, and delay distortion with and without feedback for a
single tube voice frequency amplifier.

were made on an experimental one-tube amplifier, 35-8500 cycles,
feeding back around the low side windings of the input and output
transformers.

Figure 12 gives the gain-frequency characteristic of an amplifier
with and without feedback when in the jS-circuit there was an equalizer

100

»*

WITHO

i
UT F(

lEDBAC

■^

WITH

r 1

FEE

3BA(

5000 10,000

FREQUENCY IN CYCLES PER SECOND

50.000 100.000

Fig. 12 — (iain-frequency characteristic of an amplifier with an equalizer in the
/3-circuit. This was designed to have a gain frequency characteristic with feedback
of the same shape as the loss frequency characteristic of a non-loaded telei)hone cable.

18 BELL SYSTEM TECHNICAL JOURNAL

designed to make the gain-frequency characteristic of the amplifier
with feedback of the same shape as the loss-frequency characteristic
of a non-loaded telephone cable.

Conclusion

The feedback amplifier dealt with in this paper was developed
primarily with requirements in mind for a cable carrier telephone
system, involving many amplifiers in tandem with many telephone
channels passing through each amplifier. Most of the examples of
feedback amplifier performance have naturally been drawn from
amplifiers designed for this field of operation. In this field, vacuum
tube amplifiers normally possessing good characteristics with respect
to stability and freedom from distortion are made to possess super-
latively good characteristics by application of the feedback principle.

However, certain types of amplifiers in which economy has been
secured by sacrificing performance characteristics, particularly as
regards distortion, can be made to possess improved characteristics
by the application of feedback. Discussion of these amplifiers is
beyond the scope of this paper.

open- Wire Crosstalk *

By A. G. CHAPMAN

Introduction

THE tendency of communication circuits to crosstalk from one to
another was greatly increased by the advent of telephone
repeaters and carrier current methods. Telephone repeaters multi-
plied circuit lengths many times, increased the power applied to the
wires, and at the same time made the circuits much more efficient in
transmitting crosstalk currents as well as the wanted currents. Carrier
current methods added higher ranges of frequency with consequently
increased crosstalk coupling. Program transmission service added to
the difficulties since circuits for transmitting programs to broadcasting
stations must accommodate frequency and volume ranges greater than
those required for message telephone circuits.

As these new types of circuits were developed, their application to
existing open-wire lines was attended with considerable difficulty from
the crosstalk standpoint. Severe restrictions had to be placed on the
allocation of pairs of wires for different services in order to keep the
crosstalk within tolerable bounds. In many cases the existing lines
were retransposed but, nevertheless, there were still important re-
strictions. While great reduction in crosstalk was obtained by the
transposition arrangements the crosstalk reduction was finally limited
by unavoidable irregularities in the spacing of the transposition poles
and in the spacing of the wires, including differences in wire sag.
To further improve matters it was, therefore, necessary to alter the
wire configurations so as to reduce the coupling per unit length between
the various circuits.

Recently this study of wire configurations has resulted in extensive
use of new configurations of open-wire lines in which the two wires of
a pair are placed eight inches apart instead of 12 inches, the horizontal
separation between wires of different pairs being correspondingly
increased. With these eight-inch pairs it has usually been found
desirable to discard the time-honored phantoming method of obtaining

* This paper gives a comprehensive discussion of the fundamental principles of
crosstalk between open-wire circuits and their application to the transposition design
theory and technique which have been developed over a period of years. In this
issue of the Technical Journal the first half of the paper is published. In the April
1934 issue will be the concluding part, together with an appendix entitled "Calcula-
tion of Crosstalk Coefficients."

20 BRLL SYSTEM TECHNICAL JOURNAL

additional circuits so as to make it possible to obtain a greater number
of circuits by more intensive application of carrier current methods.

It is the object of this paper to outline the fundamental principles
concerning crosstalk between open-wire circuits and recent develop-
ments in transposition design theory and technique which have led
to the latest pole line configurations and transposition designs.

To those generally interested in electrical matters it is hoped that
this paper will give an insight into the problem of keeping crosstalk in
open-wire lines within proper bounds. To those interested in crosstalk
it is hoped that the paper will give a useful review of the whole matter
and perhaps an insight into the importance of some phenomena which
do not seem to be generally appreciated.

The principles set forth in this paper will also be found of con-
siderable interest in connection with problems of control of cable
crosstalk, particularly for the high frequencies involved in carrier
transmission. It will also be recognized that use is made here of the
same general principles as are used in the calculation of effects of
impedance irregularities and echoes on repeater operation. These
general principles have also been found useful in the development of
combinations or arrays of radio antennas of the long horizontal wire
type.

The art of crosstalk control in open-wire lines has grown up as a
result of the efforts of many workers. The individual contributions
are so numerous that it has not been considered practicable in this
paper to make individual mention of them except in a few special cases.

General

In the evolution of a satisfactory transposition design technique,
complicated electrical actions must be considered and it has been
convenient to divide the total crosstalk coupling into various types,
all of which may contribute in producing crosstalk between any two
circuits in proximity. The first portion of this paper is therefore
devoted largely to an examination of the underlying principles and the
definition of some of the special terms employed, such as transverse
crosstalk, interaction crosstalk, reflection crosstalk, etc. The paper then
considers the general effect of transpositions in reducing crosstalk and
how this effect depends on the attenuation and phase change accom-
panying the transmission of communication currents. Consideration
is next given to the practical significance of and methods for deter-
mining the crosstalk coefficients which are used in calculating the
crosstalk in a short part of a parallel between two currents. The
matter of type imbalances inherent in different arrangements of trans-

OPE N- WIRE CROSS TA LK

positions and used in working from short lengths to long lengths is
discussed at length. The next section of the paper is devoted to the
efifect of constructional irregularities caused by pole spacing, wire sag,
"drop bracket" transpositions, etc. Various "non-inductive" wire
arrangements are considered. The paper closes with a general
discussion of practical transposition design methods based on the
principles previously disclosed.

Underlying Principles

The discussion under this heading will cover the general causes of
crosstalk coupling between open-wire circuits and the general types
into which it is convenient to divide the crosstalk effect. The usual
measures of crosstalk coupling will also be discussed.

Causes and Types of Crosstalk

The crosstalk coupling between open-wire pairs is due almost entirely
to the external electric and magnetic fields of the disturbing circuit.
If these fields were in some way annulled there would remain the
possibility of resistance coupling between the pairs because of leakage
from one circuit to the other by way of the crossarms and insulators,
tree branches, etc. This leakage effect is minor in a well-maintained
line. It enters as a factor in the design of open-wire transpositions
only in so far as the attenuation of the circuits is affected which
indirectly affects the crosstalk.

Figure 1 indicates cross-sections of two pairs of wires designated as
1-2 and 3-4. If pair 1-2 existed alone and if the two wires were
similar, a voltage impressed at one end of the circuit would result in

Fig. 1 — Magnetic field produced by equal and opposite currents in wires 1 and 2.

22 BELL SYSTEM TECHNICAL JOURNAL

equal and opposite currents at any point. These currents would
produce a magnetic field as indicated on the figure. If circuit 3-4
parallels 1-2 a certain amount of this magnetic flux would thread
between wires 3 and 4 and induce a voltage in circuit 3-4 which would
result in a crosstalk current in this circuit. This induced voltage is,
of course, due to the difference between the two magnetic fields set
up by the opposite directional currents in wires 1 and 2. Since wires
1 and 2 are not very far apart, the resultant field is much weaker than
if transmission over wire 1 with ground return were attempted. It is
important, therefore, that the wires of a circuit be placed as close
together as practicable and that these wires be similar in material and
gauge in order to keep the currents practically equal and opposite.

Equal and opposite charges accompany the equal and opposite
currents in wires 1 and 2. The equipotential lines of the resultant
electric field set up by the two charges are also indicated by Fig, 1.
This field will cause different potentials at the surfaces of wires 3 and
4 and this potential difference will cause a crosstalk current in circuit
3-4. As in the case of magnetic induction this current may be
minimized by close spacing and electrical similarity between the two
wires of a pair.

Calculations of crosstalk coupling must, in general, consider both
the electric and magnetic components of the electromagnetic field of
the disturbing circuit.

The exact computation of crosstalk coupling between communication
circuits is very complex.^ Approximate computations are sufficient
for transposition design. In such computations, it is convenient to
divide the total coupling into components of several general types.
In calculations of coupling of these types it is assumed that the two
wires of a circuit are similar in material and gauge. If there is any
slight dissimilarity, such as extra resistance in one wire due to a poor
joint, the effect on the crosstalk may be computed separately. The
general types of crosstalk coupling are :

1. Transverse crosstalk coupling.

la. Direct.
\h. Indirect.

2. Interaction crosstalk coupling.

A multi-wire pole line involves many circuits all mutually coupled.
In explaining the above terms, it is convenient to start with the
simple conception of but two paralleling coupled circuits; Fig. 2 A

^ The general mathematical theory is given in the Carson-Hoyt paper listed under
"Bibliography."

OPEN-WIRE CROSSTALK 23

indicates such a parallel. In calculating the crosstalk coupling
between a terminal of circuit a and a terminal of circuit b, the parallel
may be divided into a series of thin transverse slices. One such
slice of thickness d is indicated on the figure. The coupling in each
slice is calculated and, then, the total coupling between circuit terminals
due to all the slices.

In Fig. 2A circuit a is considered to be the disturber and to be
energized at the left-hand end. In the single slice indicated, a trans-
mission current will be propagated along circuit a and will cause
crosstalk currents in circuit b at both ends of the slice. In this slice,
therefore, the left-hand end of circuit a may be considered to be
coupled to the two ends of circuit b through the transmission paths
flab and fab- The path «„& is called the near-end crosstalk coupling
and the path fab is called the far-end crosstalk coupling.

The presence of a tertiary circuit, such as c of Fig. 2B, changes
both the near-end and the far-end coupling between a and b in the
transverse slice. In addition to the direct couplings «„& and fab there
are indirect couplings fiacb and/„c6 by way of circuit c.

The transverse crosstalk coupling between a terminal of a disturbing
circuit and a terminal of a disturbed circuit is defined as the coupling
between these points due to all the small couplings in all the thin
transverse slices including indirect couplings in each slice by way of
other circuits. (There are also indirect couplings involving more
than one slice and these are not included in the transverse crosstalk
coupling.)

In computations of transverse crosstalk coupling it is convenient to
distinguish between the direct and indirect components. The direct
component considers only the currents and charges in the disturbing
circuit while the indirect component takes account of certain charges
in tertiary circuits resulting from transmission over the disturbing
circuit. The tertiary circuits may be circuits used for transmission
purposes or any other circuits which can be made up of combinations
of wires on the line or of these wires and ground. If there are only
two pairs on the line as in Fig. 2A there are still tertiary circuits,
namely, the "phantom" circuit consisting of pair a as one side of the
circuit and pair b as the return and the "ghost" circuit consisting of
all four wires with ground return. In a multi-wire line many of the
tertiary circuits involve the wires of the disturbing circuit. If these
tertiary circuits did not exist the currents at any point in the two
wires of the disturbing circuit would be equal and opposite. The
presence of the tertiary circuits makes these currents unequal and it
is convenient to divide the actual currents into two components, i.e.,

BELL SYSTEM TECHNICAL JOURNAL

equal eind opposite or "balanced" currents in the two wires of the
disturbing circuit and equal currents in phase in the two wires. The
latter may be called "tertiary circuit" currents. The charges on the
two wires of the disturbing circuit may be similarly divided into
components.

TO LONG

'circuits

^Qb

■ab

TO LONG
CIRCUITS '

\ I

Hcicb

CA)

\ I

\ I.

Tacb

^ N

(B)

1 ^ ^ 1

1 1 \ 1

1 1 jr^dc 1

1 "^ ^'/^"^"^^ '

1 Tcb) |ncb 1 1

1 / ^ 1 1

V-^' N ^ 1

Fig. 2 — Transverse and interaction crosstalk.

The direct component of the transverse crosstalk coupling is defined

as that part which is due to balanced charges and currents in the

disturbing circuit.^ The indirect component is defined as that part

2 " Direct" is here used in a different sense from that used in connection with the
tej-m "direct'capacity unbalance" which was originated by Dr. G. A. Campbell and
has been much used in discussions of cable crosstalk.

(

OPEN^WIRE CROSSTALK 25

which is due to charges on tertiary circuits which arise within any
thin transverse sUce due to coupHng with the disturbing circuit in
that same sHce. This coupHng, in any sHce, causes currents as well
as charges in the tertiary circuit in that slice, but, as discussed in
detail in Appendix A, the effect of these currents in producing crosstalk
currents in the disturbed circuit is small compared with the effect of
the charges. The currents and charges in the tertiary circuits in any
thin slice due to the coupling with the disturbing circuit in that same
slice may be but a small part of the total currents and charges in the
slice. The total values are due to couplings of the tertiary circuits
with the disturbing circuit in all the slices. When the total values
are considered currents as well as charges in the tertiary circuit may
be important in causing crosstalk currents in the disturbed circuit.
To consider the total currents and charges in the tertiary circuits it is
necessary to take account of both the interaction crosstalk coupling
and the transverse crosstalk coupling between disturbing and disturbed
circuits.

The nature of interaction crosstalk coupling is indicated by Figs. 2C
and 2D which indicate two successive thin transverse slices of width
^ in a parallel between two circuits a and b and the typical tertiary
circuit c. Assuming transmission from left to right on circuit a in
Fig. 2C this circuit is coupled with c in the right-hand slice by the
near-end crosstalk coupling indicated by fiac- This coupHng causes
transmission of crosstalk current (and charge) into the left-hand part
of circuit c which has both near-end and far-end crosstalk coupling
to circuit b. Consideration of these two successive transverse slices,
therefore, introduces the two compound couplings tiacncb and Uacfcb-
There are two more of these compound couplings as indicated by
Fig. 2D. There is a far-end crosstalk coupling between circuits a and
c in the left-hand slice which combines with both near-end and far-end
couplings in the right-hand slice. The compound types of crosstalk
of Fig. 2C and 2D are called interaction crosstalk since the various
slices interact on each other in producing indirect couplings. The
interaction crosstalk coupling between a terminal of a disturbing circuit
and a terminal of a disturbed circuit is defined as the coupling between
these points due to the indirect couplings involving all possible combi-
nations of different thin transverse slices.

The distinction between indirect transverse crosstalk and interaction
crosstalk is that the former takes account of the effect of indirect
crosstalk from disturbing to tertiary to disturbed circuit in a single
thin transverse slice while the latter involves indirect crosstalk from
primary circuit to tertiary circuit in one slice, transmission along the

26 BELL SYSTEM TECHNICAL JOURNAL

tertiary circuit into another slice and then crosstalk from tertiary
circuit to disturbed circuit.

The notion that there is only transverse crosstalk within any one
"thin slice" implies that the slice thickness corresponds to a distance
along the line of only infinitesimal length. If this distance were finite
it would correspond to a series of "thin slices" having interaction
crosstalk between them. Practically, however, if the distance along
the line corresponds to a line angle of five degrees or less, the interaction
crosstalk in this length is small compared with the transverse crosstalk.
A five degree line angle corresponds to a length of about .1 mile at
25 kilocycles, .05 mile at 50 kilocycles, etc. A transposed line is
divided into short lengths or segments by the transposition poles and
the line angle of these segments is ordinarily less than five degrees at
the highest frequency for which the transposition system is suitable.
Therefore, the crosstalk coupling between such transposed circuits
may be computed on the basis of transverse crosstalk wuthin any
segment and interaction crosstalk between any two segments.

As shown by Fig. 2 the interaction effect involves the four compound
couplings:

nacflcb, nacfcb, facncb, facfcb-

The near-end crosstalk couplings Hac and Ucb of Fig. 2 are usually
much larger than the far-end couplings fac and fcb. The reason for
this, as discussed in Appendix A, is that the electric and magnetic
fields of the disturbing circuit tend to aid each other in producing
near-end crosstalk coupling such as Uac, and to oppose each other in
the case of far-end coupling such as fac For this reason the compound
coupling Hacficb is the most important and is usually the only compound
coupling which requires consideration in transposition design. Since
the path n„cncb results in a crosstalk current at the far end of the
disturbed circuit, it is in connection with far-end crosstalk between
long circuits that this matter of interaction crosstalk is important.
Far-end rather than near-end crosstalk coupling is controlling in
connection with open-wire carrier frequency systems for the reasons
explained below.

Figure 3A indicates very schematically two one-way carrier fre-
quency channels routed over two long paralleling open-wire pairs.
The boxes at the end indicate the repeaters or terminal apparatus and
the arrows on these boxes the direction of transmission of this appa-
ratus. Transmission from the left on pair a results in near-end and
far-end crosstalk into pair b, as indicated by the couplings riah and fab.
The near-end crosstalk current cannot pass to the input of the terminal

OPEN-WIRE CROSSTALK

apparatus since the latter is a one-way device. In practice, to obtain
two-way circuits each of these one-way channels is associated with
another one-way channel transmitting in the opposite direction over
the same pair of wires. These return channels utilize a different band
of carrier frequencies and the near-end crosstalk current is largely
excluded from this frequency band by selective filters. The far-end
crosstalk is, therefore, the sole consideration with such a carrier system.
Use is not made of the same carrier frequencies in both directions on a
toll line largely because of difficulties in controlling the near-end
crosstalk.

— ►

K ^

(A)

►

— a. ^^

- -\

—7

l^ab

n ab

►

— ^

'n (B) 'n

Fig. 3 — Crosstalk between two one-way carrier frequency channels.

In connection with the arrangement of Fig. 3A, there is a type of
crosstalk of considerable practical importance known as ''reflection
crosstalk.'' The theory of this is indicated by Fig. 3B which shows
the same two one-way carrier channels. Transmission from left to
right on circuit a is assumed. When the transmission current I
arrives at point B, a certain portion of it will be reflected if there is
any deviation of the input impedance of the terminal apparatus from
the characteristic impedance of circuit a. This reflected current Ir
causes a near-end crosstalk current in at point B in the disturbed
circuit. Similarly, a part of the near-end crosstalk current in at point
A in the disturbed circuit may be reflected and transmitted to point B.
Therefore, two additional crosstalk currents may result from these
two reflections and such currents can enter the terminal apparatus at
B and pass through to the output of this apparatus.

28 BELL SYSTEM TECHNICAL JOURNAL

For like circuits, like impedance mismatches and like near-end
crosstalk couplings at the two ends of the line, these two additional
far-end crosstalk currents are of equal importance. Similar reflection
effects will occur at any intermediate points in the lines having im-
pedance irregularities. Since the far-end crosstalk coupling can be
much more readily reduced by transpositions than the near-end
crosstalk coupling this reflection crosstalk effect is important in
practice. It is, therefore, necessary to carefully design the terminal
and intermediate apparatus and cables to minimize impedance mis-
matches as far as practicable.

In calculation of crosstalk coupling it is ordinarily assumed that
the two wires of a circuit are electrically similar or "balanced " (except
as regards crosstalk from other wires). This is substantially true in
practice except for accidental deviations, such as resistance differences
due to poor joints and leakage differences due to cracked insulators,
foliage, etc. Resistance differences may be of considerable practical
importance and are said to cause resistance unbalance crosstalk. The
following discussion indicates the general nature of this effect.

As discussed in connection with Fig. 1, the external field of the
disturbing circuit is minimized by the opposing effects of substantially
equal and opposite currents or charges in the two wires of the circuit.
The two wires may be considered as two separate circuits, each having
its return in the ground. At any point in the line these two wires
would normally have practically equal and opposite voltages with
respect to ground. These voltages would normally cause almost equal
and opposite currents in the two wires. If the resistance of one wire
is increased due to a bad joint, the current in that wire is reduced and
the currents in the two wires are no longer equal and opposite. The
external field of the two wires and the resulting voltage induced in the
disturbed circuit are, therefore, altered. If this voltage had previously
been practically cancelled out by means of transpositions, the alter-
ation in the field would increase the crosstalk current at the terminal
of the disturbed circuit.

A resistance unbalance in the disturbed circuit will have a similar
effect as indicated by Fig. 4A. This figure shows a short length d of
two long paralleling circuits. Equal and opposite transmission cur-
rents in the disturbing circuit 1-2 are indiceited by /. Equal crosstalk
currents in the two wires of the disturbed circuit 3-4 at one end of the
short length are indicated by i. It is assumed that these crosstalk
currents have been made substantially equal by transpositions in other
parts of the line. Since the currents in wires 3 and 4 are equal and
in the same direction, there will be no current in a receiver connected

OPEN-WIRE CROSSTALK

at the terminal of the Hne between these wires. If, however, one
wire has a bad joint, the two crosstalk currents become unequal and
there will be a current in such a receiver.

Resistance unbalance crosstalk is of particular importance if two
pairs are used to create a phantom circuit in order to obtain three
transmission circuits from the four wires. The distribution of the
phantom transmission current I p in a short length of the two pairs is
indicated by Fig. 4B. Ideally, half the phantom current flows in
each of the four wires. The two currents in wires 1 and 2 are then
equal and in the same direction and there will be no current in terminal
apparatus connected between wires 1 and 2. In other words, trans-
mission over the phantom circuit results in no crosstalk in the side
circuit 1-2. The same may be said of side circuit 3-4. A bad joint
in any wire, such as 3, makes the two currents in wires 3 and 4 unequal
and results in a current in the side circuit 3-4.

d -

1 [

_[p^^2

2 1

TO LONG
CIRCUITS

_A A A . 1

TO LONG — ^
CIRCUITS

3 !

V Vv^
R

IpJrZ

4 1

PHANTOM CURRENT = Ip
(A) CB)

Fig. 4 — Effect of resistance unbalance on crosstalk.

The phantom-to-slde crosstalk effect of resistance unbalance is
much more severe than the effect on crosstalk between two side
circuits or two non-phantomed circuits. The reason for this is evident
from Figs. 4B and 4A. In Fig. 4B the entire transmission current of
the disturbing phantom circuit normally flows in the two wires of the
disturbed side circuit and if a resistance unbalance causes a small
percentage difference in the currents in these two wires objectionable
crosstalk results. In Fig. 4A only crosstalk currents flow in wires 3
and 4 and a much larger percentage difference between these small
currents can be tolerated.

In designing and operating phantom circuits, it is necessary to
exercise great care to minimize any dissimilarity between the two wires
of a side circuit, in order to avoid crosstalk from a phantom to its
side circuit or vice versa. Otherwise, the problem of crosstalk between

BELL SYSTEM TECHNICAL JOURNAL

a phantom circuit and some other circuit is generally similar to the
problem of crosstalk between two pairs. In other words, the discussion
of transverse, interaction and reflection crosstalk is applicable.

Measures of Crosstalk Coupling

In designing transposition systems, the usual measure of the coupHng
effect between two open-wire circuits is the ratio of current at the
output terminal of the disturbed circuit to current at the input terminal
of the disturbing circuit. For circuits of different characteristic
impedances this current ratio must be corrected for the difference in
impedance. The corrected current ratio is the square root of the
corresponding power ratio.

The current ratio is ordinarily very small and for convenience is
multiplied by 1,000,000 and called the crosstalk coupling or, in brief,
the crosstalk. This usage will be followed from this point in this
paper. For example, crosstalk of 1000 units means a current ratio
of .001. Crosstalk may also be expressed as the transmission loss in
db corresponding to the current ratio. A ratio of .001 means a
transmission loss of 60 db corresponding to 1000 crosstalk units.

Ir I

TO LONG CIRCUITS

Fig. 5 — Schematic of near-end and far-end crosstalk.

Figure 5 indicates two paralleling communication circuits a and b
with an e.m.f . impressed at one end of circuit a. The crosstalk currents
in and if in circuit b are due to the crosstalk coupling in length AB.
The near-end crosstalk in the length AB \s the ratio X^HuJIa, while the
far-end crosstalk is IOH/JIa- The ratio 10H//Ib has been called the
"output-to-output" or "measured" crosstalk. This ratio is a con-
venient measure of far-end crosstalk between parts of similar circuits
because it is related in a simple way to the far-end crosstalk between
the terminals of the complete circuits. The following discussion
explains this relation.

OPEN-WIRE CROSSTALK 31

Both of the currents Ib and if will be propagated to point C. They
will be attenuated or amplified alike if the circuits are similar and their
ratio will be unchanged. The output-to-output crosstalk at C due
to the length AB will, therefore, be the same as that determined for
point B. In other words lOH/JIc will equal IOH/JIb. The far-end
crosstalk between the terminals A and C, due to length AB, will be
IOH/JIa' This differs from the output-to-output crosstalk at C in
that the reference current is Ia instead of Ic. The part of the far-end
crosstalk between A and C due to AB is, therefore, obtained from the
output-to-output crosstalk at B by simply multiplying by the attenu-
ation ratio Ic/Ia- If the output-to-output crosstalk is expressed as
a loss in decibels, the far-end crosstalk is obtained by adding the net
loss of the complete circuit between A and C.

Effects of Transpositions

The eflfects of transpositions on both the transmission currents and
the crosstalk currents will now be discussed in a general way. The
general method of computing the crosstalk between circuits without
constructional irregularities and transposed in any manner will also
be outlined.

General Principles

If there is only one circuit on a pole line, and this is balanced and
free from irregularities, the communication currents will be propagated
along this circuit according to the simple exponential law. If a
current is propagated from the start of the circuit to some other point
at a distance L, the magnitude of the current will be reduced by the
attenuation factor e~"^ and the phase of the current will be retarded
by the angle /SL where a is the attenuation constant and (3 is the
phase change constant.

If there are a number of circuits on a pole line this simple law of
propagation may be altered due to crosstalk into surrounding circuits.
This is illustrated by the curves. Fig. 6, which indicate the relation
between observed output-to-input current ratio and frequency for two
different circuits, each about 300 miles long and having 165-mil copper
wires. The number of decibels corresponding to the current ratio is
plotted rather than the ratio itself. For the simple law of propagation
such curves would show the number of decibels increasing smoothly
with frequency due to increasing losses in the line wires and insulators.
The upper curve is for a circuit too infrequently transposed for the
frequency range covered and the current ratio is abnormally small
at particular frequencies. The corresponding number of decibels is
abnormally large. The lower curve is for a circuit much more fre-

M BELL SYSTEM TECHNICAL JOURNAL

quently transposed and its current ratios practically follow the simple
propagation law mentioned above over the frequency range shown.

Even though a circuit is very frequently transposed, its propagation
constant is slightly affected by the presence of other circuits on the line.
This may be explained by consideration of Figs. 2B and 7. As previ-
ously explained, Fig. 2B indicates the indirect transverse crosstalk by
way of a tertiary circuit in one thin transverse slice of a parallel

BEFORE / \

retransposing/ \

^-"^

■/-^/

/ AFTER

'retransposing

__«/0'^

/^-

0 5 10 15 20 25 30

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 6 — Effect of transpositions on attenuation of an open-wire pair.

between two long circuits a and h. The circuit c has currents and
charges due to crosstalk from the disturbing circuit a. These currents
and charges not only alter the crosstalk currents in circuit h but also
react to change the transmission current in circuit a. Since circuits
a and c are loosely coupled, this reaction effect could usually be esti-
mated with sufficient accuracy by calculating the crosstalk from a to
c and back again and neglecting the further reactions of the change
in the current in a on the current in c, etc.

OPEN-WIRE CROSSTALK

Figure 7 shows the crosstalk paths from a to c and back again.
In this figure, circuit a is indicated as two separate circuits for com-
parison with Fig. 2B. It is assumed that circuit a in Fig. 7 is energized
at point A, the currents J a and Ib being the currents which would
exist at the input and output of the short length d if there were no
tertiary circuits. The near-end crosstalk path indicated by n will
cause a small crosstalk current in at point A in circuit a. There will
be a crosstalk path similar to n in each thin slice of the parallel between
a and c. Each of these paths will transmit a small crosstalk current
to point A in circuit a. The sum of all these crosstalk currents will
increase the input current Ia and, therefore, the impedance of circuit

r TO LONG

CIRCUITS"

Fig. 7 — Effect of circuit c on propagation in circuit a.

a is lowered. Thin slices remote from the sending end will contribute
little to this effect, since the crosstalk currents from such slices will be
attenuated to negligible proportions. A long circuit on a multi-wire
line will, therefore, have a definite sending-end impedance slightly
lower than that for one circuit alone on the line.

Figure 7 also indicates a far-end crosstalk path / which produces a
crosstalk current if at point B in circuit a. This reduces the trans-
mission current Ib at this point and, therefore, increases the attenu-
ation constant of the circuit. For calculations of both the circuit

34 BELL SYSTEM TECHNICAL JOURNAL

impedance and attenuation, the effect of surrounding circuits is taken
care of in practice by using a capacity per unit length sHghtly higher
than the value which would exist with only one circuit on the line.
The proper capacity to use is determined in practice by measurements
on a short length of a multi-wire line.

The effect on the propagation constant of the transverse crosstalk
paths indicated by n and / of Fig. 7 cannot be suppressed by trans-
positions. As explained later, if the two circuits marked a were
actually different circuits, the effect could be largely suppressed by
transposing one circuit at certain points and leaving the other circuit
untransposed at these points. Since the disturbing and disturbed
circuits indicated by Fig. 7 are actually the same circuit, they must
be transposed at the same points and, therefore, the transverse cross-
talk effect cannot be suppressed by frequent transpositions.

Figure 7 also shows a crosstalk path marked r. This is one of the
possible interaction crosstalk paths. The effect of such paths on the
impedance and attenuation of the circuit may be largely suppressed
by suitable transpositions. The difference between the two curves of
Fig. 6 is due to lack of this suppression in the case of the upper curve.

Such an extreme effect of crosstalk reacting back into the primary
or initiating transmission circuit and thus affecting direct transmission
is seldom important in practical transposition design. A marked
reaction on the primary circuit would necessitate such large crosstalk
currents in neighboring communication circuits as to make them unfit
for communication service at the frequency transmitted over the
primary circuit. Therefore, it is only when the neighboring circuits
are not to be used at this frequency that transposition design to control
simply the direct transmission becomes of practical importance.
When many circuits on a line are used for carrier operation, the
crosstalk currents must be made so weak (by transpositions, physical
separation of circuits, etc.) that their reactions back into the primary
circuits are very small.

The effect of transpositions on crosstalk from one circuit into another
different circuit will now be considered. The discussion of the control
of this effect is the main object of this paper.

Figure 8A shows a short segment of a parallel between two long
circuits and a near-end crosstalk coupling marked n. The segment
could be divided into a series of thin slices and theoretically there
would be interaction crosstalk between different slices. The segment
length is, however, assumed to be short enough to neglect interaction
crosstalk. The coupling n is, therefore, due either to direct or indirect
transverse crosstalk in the short segment or to both of these types of

OPEN-WIRE CROSSTALK

crosstalk. If circuit a is energized from the left, a near-end crosstalk
current in results at point A in circuit b.

If two successive short segments are considered, as indicated by
Fig. 8B, there will be a near-end crosstalk coupling n in each segment
and each of these couplings will result in a crosstalk current at point A

RESULTANT

h- — d— t- — d —1

i-*lA |-*Ib I

k k 1

1 \ 1 \ 1

1 ri 1 n 1
1^1 ^ 1

k--^^ i

(B)

1— Ia I Ib*-I

i-- X -:

1 b 1 1

-Is

RESULTANT

f JA.

(D)

Fig. 8 — Effect of transpositions on transverse crosstalk.

of circuit b. This is indicated by the vector diagram over the figure,
where in indicates the crosstalk current due to the segment AB and
in indicates the crosstalk current at A due to BC. The latter current
is slightly smaller and slightly retarded in phase with respect to in
because in order for i,/ to appear at point A, the transmission current
Ia must be propagated a distance d and the resulting crosstalk current

36 BELL SYSTEM TECHNICAL JOURNAL

at B must also be propagated a distance d in order to reach A. As
indicated by this vector diagram the total crosstalk current due to the
two short segments is a little less than the arithmetic sum of the
individual crosstalk currents.

Figure 8C is like Fig. 8B except that a transposition is inserted in
the middle of circuit a at point B. This reverses the phase of the
transmission current at the right of B and also reverses any crosstalk
current due to current in circuit a between B and C. As a result the
crosstalk current in of Fig. 8B is reversed and the resultant of the two
crosstalk currents is very much reduced as indicated by the vector
diagram of Fig. 8C. The angle between i„ and in is proportional to
the length 2d which equals A C. The tendency for the two currents to
cancel may, therefore, be increased by reducing the length AC which,
in a long line, would mean increasing the number of transpositions.

Figure 8D is like Fig. 8B except that the far-end transverse crosstalk
coupling / in each of the two short segments is considered. The
coupling in the left-hand segment results in a crosstalk current at
point B of circuit h, w^hich is propagated to point C as indicated by if.
The far-end crosstalk coupling in the right-hand segment produces a
crosstalk current i/ at point C. Since the total propagation distance
is from ^ to C for both of these crosstalk currents, they must be equal
in magnitude and in phase if circuits a and h are similar. This is
indicated by the vector diagram of Fig. 8D. A transposition at point
B in either circuit would reverse one of these crosstalk currents and,
therefore, the resultant crosstalk current would be nil.

From consideration of Figs. 8C and 8D, it may be seen that if both
circuits were transposed at point B, the sum of the crosstalk currents
for the two segments would be the same as if neither circuit were
transposed. Transposing one circuit reverses the phase of one of
the component crosstalk currents, but if the second circuit is also
transposed the original phase relations between the two currents are
restored.

The foregoing discussion applies only to transverse crosstalk as
discussed in connection with Fig. 2. When interaction crosstalk must
be considered, a different principle is involved.

In connection with Fig. 8D, it w^as shown that the transverse far-end
crosstalk between similar circuits could be readily annulled by trans-
posing one of the circuits at the center of their paralleling length.
Far-end crosstalk of the interaction type is not so readily annulled.
The effect of transpositions on this type of crosstalk is indicated by
Fig. 9.

This figure shows four short segments in a parallel between two

OPEN-WIRE CROSSTALK

circuits a and h, there being an interposed tertiary circuit c. Inter-
action crosstalk involving two near-end crosstalk couplings is con-
sidered since this is usually the controlling type. There is an inter-
action crosstalk path designated r between the first two segments as
indicated by Fig. 9A. There is a similar path between the third and
fourth segments. Each of these paths would produce a far-end
crosstalk current in circuit h at point E. For similar circuits these
currents would be equal in magnitude and would add directly. The
two currents can be made to cancel by transposing one of the circuits
at C, the midpoint of the parallel. Such a transposition also cancels
the transverse far-end crosstalk in length A C against that in length CE.
There remains, however, the interaction crosstalk between length CE
and length A C.

— ° —

(A)

CB)

Fig. 9 — Effect of transpositions on interaction crosstalk.

Figure 9B shows a transposition at C in circuit a and also other
transpositions whose purpose is to minimize the interaction crosstalk
between length CE and length AC. This crosstalk coupling, desig-
nated by r', is a compound effect, depending on the near-end crosstalk
between circuit a and circuit c in length CE and the near-end crosstalk
between c and b in length AC. The near-end crosstalk coupling
between a and c in length CE can be greatly reduced by a transposition
in circuit a at point D, while the crosstalk coupling between c and b in
length A C can likewise be reduced by a transposition at point B in
circuit b. The latter two transpositions would not, however, minimize
the interaction crosstalk between CE and AC with circuit b as the

BELL SYSTEM TECHNICAL JOURNAL

disturbing: circuit and it is necessary, therefore, to transpose both
circuits at points B and D. The addition of these four transpositions
does not afifect the cancellation of far-end crosstalk in length AC
against that in length CE by means of the transposition at C. After
the four transpositions are added, length AC is still similar to length
CE and the far-end crosstalk currents at E, due to these two lengths,
are equal. Therefore, they will cancel when one of them is reversed
in phase by the transposition at C.

It may be concluded that, while transposing both circuits at the
same points has no effect on transverse crosstalk, it has a large effect
on the interaction crosstalk. An experimental illustration is given in
Fig. 10. This figure shows frequency plotted against output-to-output

I 1 1 1 1 1 1 IDISTURBING
1 1 1 1 1 1 1 1 CIRCUIT

] 1 j 1 1 ] ] DISTURBED
1 1 1 1 1 1 1 1 CIRCUIT

X T X T X

REGULAR TRANSPOSITION POLE;

TXT
■■ EXTRA TRANSPOSITION POLE

5 10 15 20 25 30

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 10 — Effect on far-end crosstalk of e.xtra transpositions in both circuits.

far-end crosstalk between the two side circuits of a phantom group on
a 140-mile length of line. The curve marked A is for the two circuits
transposed for voice-frequency operation. Curve B is for the two
circuits transposed in the same manner except that four transpositions

OPEN-WIRE CROSSTALK 39

per mile were added to both circuits at the same points which are
indicated by x on Fig. 10. The large effect of these transpositions
shows the practical importance of the interaction type of far-end
crosstalk.

In connection with Fig. 9B, there arises the question of how far
apart the transpositions can be placed without serious crosstalk, in
other words, how long is it permissible to make the segment d. If
this length is increased the transpositions at B and .0 become less
effective in suppressing the near-end crosstalk between a and c in
length CE and between c and h in length AC. The degree to which
the interaction crosstalk path r' must be suppressed is, therefore,
important in determining the maximum permissible length of d. If d
is increased the transposition at C becomes less effective in controlling
the near-end crosstalk between a and h and, therefore, the length d
also depends on the permissible near-end crosstalk.

It may be noted that transpositions at B and D in but, one of the
circuits a or h will help to suppress r' , but the suppression is less
effective than if both circuits are transposed at these points. If a is
transposed at B and D the near-end crosstalk between a and c in
length CE is reduced but the near-end crosstalk between c and h in
length AC IS not reduced. The product of these two near-end crosstalk
values is greater, therefore, than if they had both been reduced by
transposing both circuits at B and D.

Crosstalk Coefficients

The crosstalk between any two long open-wire circuits may be
calculated by dividing the parallel into a succession of thin transverse
slices and summing up the crosstalk for all these slices. To calculate
the crosstalk in any slice it is necessary to know certain "crosstalk
coefficients." The discussion below defines these coefficients and
describes briefly how they are measured or computed.

Figures 2 A and 2B indicate both near-end and far-end crosstalk
coupling of both the direct and indirect transverse types in a thin
transverse slice. Any of these couplings may be expressed in crosstalk
units and the value of the coupling in a short length divided by the
length in miles is called the crosstalk per mile. Since, as shown in
the previous section, the crosstalk may not increase directly as length,
strictly speaking, the crosstalk per mile is the limit of the ratio of
coupling to length as the length approaches zero. The crosstalk per
mile includes both the direct and indirect types of transverse crosstalk
coupling. In the frequency range of interest (i.e., above a few hundred
cycles for near-end crosstalk and above a few thousand cycles for

40 BELL SYSTEM TECHNICAL JOURNAL

far-end crosstalk) this total transverse coupling varies about directly
with the frequency and the crosstalk coefficient commonly used is the
crosstalk per mile per kilocycle.

If many wires are involved, it is impracticable to determine these
coefficients with good accuracy by computation and they are, therefore,
derived from measurements. Examples of near-end and far-end
coefficients, plotted against frequency, are shown in Fig. 11. The
coefficients are for pairs designated 1-2 and 3-4 on the pole head
diagram shown on the figure. These coefficients were derived from
measurements of the near-end and far-end crosstalk over a range of
frequencies. The length of line was about .2 mile and, for the range
of frequencies covered, this length is sufficiently short so that inter-
action crosstalk is negligible and the transverse crosstalk is directly
proportional to the length. The coefficients plotted are, therefore,
nearly equal to the measured values of crosstalk divided by the length
and by the frequency. (A small correction was made at the higher
frequencies to allow for deviation of near-end crosstalk from simple
proportionality to length and the curves were "smoothed" through
the actual points calculated from the measurements.)

In order to obtain the crosstalk coefficients applicable to a short
part of a long line, all the wires on the line were terminated in such a
manner as to roughly simulate their extension for long distances in
both directions, but without crosstalk coupling between the test pairs
in such extensions. This is done by terminating each pair at each
end with a resistance approximating its characteristic impedance and
connecting the midpoint of each resistance to ground through a second
resistance. These latter resistances terminate any phantom of two
pairs as indicated on Fig. 11 for pairs 1-2 and 3-4. Any circuit with
ground return is also terminated by these resistances.

Both of the test pairs are transposed at the midpoint of the line
during the measurement. This minimizes the currents reaching the
ends of the tertiary circuits and makes even the above approximate
termination of the tertiary circuits of little importance.

Figure 11 shows near-end and far-end crosstalk coefficients for three
conditions, A , B, and C. The two curves marked A show the measured
values with all wires terminated and the test pairs transposed as
described above.

For curves B, only the transposed test pairs were terminated as
described above and the other wires were opened at the middle, at the
quarter points and at both ends. Since no section of any of these
wires connected points of substantially different potential in the field
of the disturbing circuit there were practically no currents or charges

OPEN-WIRE CROSSTALK

in these wires and the crosstalk coefficients for the two test pairs were
practically the same as if the other wires had been removed from the
line. It will be seen that the crosstalk coefficients for curves B are

OSCILLATOR

8r-26--i8r- 28 — H8r^26-H8

-iH B p- ^b -»H o r"—

— H 8 h«-26-H 8 l-»-

1 T

1 2

t\J

1 1

■^

'I ^

■=1

^AA^

WIRE CONFIGURATION

A
B

- — - __

— ^

0 5 10 15 20 25 30 35 40 45 50

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 11 — Near-end and far-end crosstalk coefficients between pairs 1-2 and 3-4.

less than those for curves A. The coefficients of curves B involve
tertiary circuits, however, since there could be crosstalk currents in
the phantom of the two test pairs and also in the ghost circuit involving
wires 1 to 4 with ground return.

42 BELL SYSTEM TECHNICAL JOURNAL

Curves C show the coefficients with the test pairs without transpo-
sitions and terminated at both ends as accurately as practicable, but
without the midpoints of these terminations connected to ground to
terminate the phantom and ghost circuits. These tertiary circuits
were, with this arrangement, prevented from connecting points of
substantially different potential and the coefficients of curves C,
therefore, approach the direct crosstalk coefficients. It is extremely
difficult to experimentally determine the direct far-end coefficient.
It may be computed, however, and the computed value which assumes
perfect terminations and the effect of the phantom completely removed
is shown by curve C.

It may be noted that the near-end crosstalk coefficients are about
independent of frequency. This is ordinarily true above a few
hundred cycles. The total far-end coefficient (curve A) is about
independent of frequency in the important carrier frequency range.
The direct far-end coefficient of curve C decreases considerably with
frequency for reasons discussed in Appendix A. Since transpositions
are ordinarily designed for the condition of a number of wires on a line,
the total crosstalk coefficient is the one usually used in practice.

Curves C of Fig. 11 also indicate that the direct near-end coefficient
is much larger than the direct far-end coefficient. This is usually
true and, as discussed in detail in Appendix A, the explanation is that
the crosstalk currents caused by the electric and magnetic fields add
almost directly in the case of direct near-end crosstalk but tend to
cancel in the case of direct far-end crosstalk. As discussed in the
appendix, the indirect (vector difference of curves A and C) crosstalk
in a very short length is due almost entirely to the electric field of the
tertiary circuits and is the same for both near-end and far-end crosstalk.
In Fig. 11, the total near-end coefficient (curve A) is increased by the
indirect crosstalk since curve C is lower than curve A. The reverse
is usually true, however. In the case of far-end crosstalk the total
coefficient is usually increased by the indirect crosstalk.

Crosstalk coefficients are vector quantities and may be measured in
magnitude and phase. If it is desired to compute the crosstalk
between two long pairs of wires which do not change their pin positions,
it is only necessary to know the magnitude of the crosstalk coefficient,
since the problem is to determine the ratio of the crosstalk for many
elementary lengths to the crosstalk for one such length. However,
if it is desired to know the crosstalk between long circuits which do
change their pin positions, several crosstalk coefficients must be known,
one for each combination of pin positions. In order to determine the
total crosstalk for several segments of a line involving different pin

OPEN-WIRE CROSSTALK 43

positions, it is necessary to know both the phase and magnitude of
the crosstalk coefficients. For practical purposes, however, the
coefficients may, in most cases, be regarded as algebraic quantities
having sign but not angle.

The direct component of the total crosstalk coefficient may be
readily computed as discussed in Appendix A. If more than a very
few wires are involved, an exact calculation of the indirect component
is impracticable but a fair approximation may be obtained by the
method discussed in Appendix A. This method is used when a wire
configuration is under consideration but is not available for measure-
ment.

As pointed out in 1907 by Dr. G. A. Campbell, an accurate calcu-
lation of the total crosstalk coefficient would involve determination
of the "direct capacitances" between wires of the test pairs. Since
these capacitances are functions of the distances between all combina-
tions of wires on the lead and between wires and ground, their calcu-
lation is usually impracticable. In the past, the crosstalk coefficients
were computed by a method proposed by Dr. Campbell which involved
measurement of the direct capacitances.^

The part of the coefficient due to the electric field was computed
from the "direct capacitance unbalance." The part due to the mag-
netic field was computed as discussed in Appendix A. When loaded
open-wire circuits were in vogue it was necessary to be able to separate
the electric and magnetic components of the coefficients. After
loading was abandoned this separation was unnecessary and it was
found more convenient to measure the total coefficients than to
measure the direct capacitances or dilTerences between pairs of these
capacitances.

As previously discussed, in designing transpositions it is necessary
to compute the interaction type of crosstalk indicated by Fig. 2C, and
it is, therefore, necessary to have some coupling factor for use in this
computation. Such a coupling factor could, theoretically, be deter-
mined as indicated schematically by Fig. 12. The interaction crosstalk
between two short lengths of line would be measured by transmitting
on one pair and receiving on the other pair at the junction of the two
short lengths as indicated by the figure.

If there were but a single tertiary circuit such as c of the figure,
the crosstalk measured would be that due to the compound crosstalk
path fiacncb- In this product, Uac is the near-end crosstalk between a
and c in the right-hand short length d and rich is the near-end crosstalk
between c and h in the left-hand short length. Since nac and rich when

^See papers by Dr. Campbell and Dr. Osborne listed under "Bibliography."

BELL SYSTEM TECHNICAL JOURNAL

expressed in crosstalk units are current ratios times a million, their
product nacficb is a current ratio times a million squared. The crosstalk
measured would be this current ratio times a million or WacWc!,10"^.

For small values of d, riac and Ucb vary directly as the frequency and
as the length d. Therefore:

neb = NcbKd,

WacWcf-lO-^ = NacNcbKHnQi-\

where Nac and Neb are the near-end crosstalk coefficients, K is the
frequency in kilocycles and d is expressed in miles. The measured

TO TERMINATIONS
REPRESENTING
LONG CIRCUITS

DETECTOR

Fig. 12 — Theoretical method of measuring interaction crosstalk coefficient.

crosstalk WacWcblQ-^ divided by KW gives the quantity iVaciVcblO"''
which may be designated as /«& and called the interaction crosstalk
coefficient. Values of lab determined from crosstalk measurements on
multi-wire lines would include the efifect of numerous tertiary circuits
instead of that of a single tertiary circuit as indicated by Fig. 12.

While the interaction crosstalk coefficient hb could theoretically be
measured as outlined above, it is simpler to deduce an approximate
value from the measured value of the far-end crosstalk coefficient Fab-
The indirect component of Fab is due to the tertiary circuits and must,
therefore, be related to lab which is also due to these circuits. As
discussed in detail in Appendix A:

lab — —

approximately.

OPEN-WIRE CROSSTALK 45

In this expression K is the frequency in kilocycles and Tc = «c + j^c
is the propagation constant of the tertiary circuit c. On a multi-wire
line there would be numerous tertiary circuits with various values of 7.
With practicable wire sizes the attenuation constants indicated by a
are small compared with the phase change constants indicated by /3.
Measurements of crosstalk indicate that the values of jS are all in
the neighborhood of the value given by the expression irK/90. This
corresponds to a speed of propagation of 180,000 miles per second
which is about the average for the present carrier frequency range.
Neglecting the attenuation constants :

.- .tK
Ic =JI3 =J-9Q-.

T — — • ^Tr/^gj,
~ -^ 90 '

This relation is much used in transposition design. As noted above,
the indirect component of Fab should, strictly speaking, be used to
obtain lab- In most cases, however, the total value of Fab may be
used since this total is determined largely by the indirect component.

Type Unbalance

A conception important in transposition design is that of "type
unbalance." This conception will now be explained and the general
method of computation will be discussed.

As we have seen, any two open-wire circuits tend to crosstalk into
each other due to coupling between them. By transposing the circuits,
the coupling in any short length of line is nearly balanced in another
short length by a second coupling of about the same size but about
opposite in phase. This balancing is never perfect and there is always
a residual unbalanced coupling due to (1) attenuation and change in
phase of the disturbing transmission current and resulting crosstalk
currents as they are propagated along the circuits and (2) irregularities
in the spacing of the transpositions and irregularities in the spacings
between the various wires. The term "type unbalance" has been
chosen to indicate the residual unbalance caused by propagation
effects. It is expressed as an "equivalent untransposed length," that
is, the type unbalance times the crosstalk per mile gives the residual
crosstalk due to propagation effects assuming no constructional
irregularities.

The method of computing the type unbalance for near-end crosstalk
will now be discussed. The part of the near-end crosstalk due to
interaction between all the different thin slices of line may be ignored

BELL SYSTEM TECHNICAL JOURNAL

since, as discussed in connection with Figs. 2C and 2D, the interaction
crosstalk involves the product of a near-end crosstalk path and a far-
end crosstalk path. This product is small since the coupling through
the far-end path is inherently small. Therefore, the interaction
crosstalk coefficient is much smaller for near-end crosstalk than for
far-end crosstalk, while for the transverse crosstalk coefficients the
reverse is true.

As was indicated by the discussion of Fig. 8B, the transverse near-
end crosstalk between two long circuits may be computed by dividing
the parallel into short segments, each having the same transverse
crosstalk coupling. The coupling between circuit terminals for any
segment will be different from that at the segment terminals due to
propagation effects as explained in connection with Fig. 8B. There-
fore, the coupling at the circuit terminal for each segment must be
determined and, finally, the sum of the coupling values for all the
segments.

The simplest case is that of two non-transposed circuits. The
problem is indicated by Fig. 13 which is like Fig. 8B except that
more segments are showm.

.d^

--f

-4

j_.

Fig. 13^ Method of computing near-end crosstalk between untransposed
circuits in length D.

The near-end crosstalk coupling n at point A due to the first segment
is NKd, where N is the crosstalk coefficient and K is the frequency in
kilocycles. The crosstalk current from the second segment relative to
that from the first segment is attenuated by the factor e-("i+"!)'^, and
also retarded in phase by the angle e~''^^i+''2^'^. In other words, the
crosstalk current from the second segment is equal to the crosstalk
current from the first segment times the factor e~('''i+'^2''^, where 7x and
72 are the propagation constants for the two circuits and y equals
a -\-j^. Letting 7 be the average propagation constant, the coupling

OPEN-WIRE CROSSTALK 47

at point A for the second segment is equal to that for the first segment
times e"^'*"^ or NKde~^'^'^. The coupling at point A for the third seg-
ment is NKde'*^'^. The sum of the crosstalk couplings at point A at
all the segments is, therefore:

NKd{l + e-^y^ + e-^y^ + e-^yi + etc.).

This expression may be summed up for the number of segments
corresponding to the total length D. It is simpler, however, to let
d be an infinitesimal length and to integrate over the length D, i.e.,
from point A to point B of Fig. 13. This gives for the total near-end
crosstalk for non-transposed circuits:

1 _ ,-270
NK^—^ .

In the special case when D is only the usual short segment between
transposition poles, the above expression is practically equal to NKD.
The near-end crosstalk between circuits having transposition poles
spaced a considerable distance D apart may now be computed. Figure
14 shows a length 2D in a parallel between two long circuits, there
being a transposition in one circuit at the center of 2D. The near-end
crosstalk for the length AB is given by the above expression. The
near-end crosstalk at point A for the length BC will be the same
expression multiplied by the propagation factor e~^y^ and reversed in
sign due to the effect of the transposition. The near-end crosstalk at
point A for the length 2D will, therefore, be the sum of the values for
lengths AB and BC. This sum is:

1 _ .-270
NK—^ (1 - e-2^^).

This quantity divided by NK is the type unbalance for the length
2D of Fig. 14. If D is only the length of a short segment the above
expression is about equal to NKD{2yD).

Similarly the near-end crosstalk at point A for a length 2>D will be:

NK- — ^ (1 - e-2^^ T 6-4^^°)

and the type unbalance is this quantity divided by NK. For a
length 4Z) the quantity in the parentheses becomes (1 — e"^''^^ T tr'^'^^
1= fT^''^), etc. The sign of each term in the parentheses is determined
by the arrangement of "relative" transpositions, i.e., those at points
where only one of the two circuits is transposed. Each term corre-

BELL SYSTEM TECHNICAL JOURNAL

sponds to a length D. The transposition at the start of the second
length (at point B of Fig. 14) reverses the sign of the term for the
second length and also the signs for the following lengths until another
transposition is reached which makes the next sign plus, etc.

A practical open-wire line is divided into a series of "transposition
sections" of eight miles or less. In each section the crosstalk between
any two circuits is approximately balanced out by means of trans-
positions. A main purpose of this division ipto sections is to provide
suitable points for circuits to drop off the line. A circuit on the line
for a part of a section may have more crosstalk to a through circuit
than if the parallel extended for the whole section since coupling in

U D -i- D

Fig. 14 — -Near-end crosstalk in length 2D between circuits a and b with circuit a

transposed in the middle.

the last part of the section may tend to subtract from the coupling in
the first part. The ends of sections are, therefore, the most suitable
points for circuits to leave or enter the line. Ideally, the sections in a
line should all be alike as regards length and transposition arrange-
ments since this makes it practicable to so design the transpositions
that residual crosstalk in one section tends to cancel that in another
section. Practically, the sections vary in length and, therefore, in the
transposition arrangements because the ends of some of the sections
must fall at particular "points of discontinuity" determined by
branching circuits and by requirements for balance against induction
from power circuits.

In designing the transposition sections, type unbalances are com-
puted for the section lengths of eight miles or less. For such lengths,
the general method of computing type unbalances may be simplified.
The general method involves the vector propagation constant 7. For
a length as short as a single transposition section, attenuation can,
ordinarily, be neglected. Therefore, in the type unbalance formulas
7 can be replaced by 7/3 which greatly simplifies the computations.

OPEN-WIRE CROSSTALK 49

Since attenuation can be neglected, the type unbalance for a trans-
position section depends only on the line angle ^D. Since j8 increases
practically directly with frequency, a plot of type unbalance against
/3Z> indicates the variation of type unbalance with frequency for a
fixed length or the variation with length for a fixed frequency. It is
convenient to plot the product of type unbalance and frequency (in
kilocycles) since this product multiplied by the crosstalk coefficient
gives the crosstalk. Two such plots for near-end type unbalance
times frequency are shown on Fig. 15A. The plot marked P is for the
condition of two circuits non-transposed or transposed alike. The
plot marked 0 is for the same arrangement except for one relative
transposition at the midpoint of the parallel.^ The figure has a
frequency scale corresponding to a length of eight miles as well as the
general ^D scale in degrees.

It will be seen that, for the case of no relative transpositions, the
crosstalk varies directly with the frequency for only a short distance
at the start of the curve. The effect of one relative transposition is to
greatly reduce the crosstalk for small values of j8L. For larger values
the crosstalk is increased. It may be noted that the minimum values
shown on the curves are somewhat in error since attenuation was
neglected.

The minimum values in the P curve are due to "natural transposi-
tions" in the non-transposed circuits. When the line angle is 180
degrees the crosstalk at the near-end of the disturbed circuit due to
the second half of the line is just 180 degrees out of phase with the
crosstalk due to the first half. This reversal in phase is due to the
phase change accompanying the propagation of current to the mid-
point and back. The total crosstalk due to both halves of the line
lengths is the same as if the crosstalk coupling in the second half were
translated to the near-end and the parallel without phase change but
one circuit was transposed at the mid-point. When the line angle is
360 degrees the "natural transpositions" are at the quarter points, etc.

The near-end crosstalk between any two circuits in a transposition
section may be estimated by multiplying the crosstalk coefficient by
values of type unbalance times frequency similar to those of Fig. 15 A.
The total crosstalk in a succession of similar transposition sections is
calculated at any particular frequency by working out a factor similar
to the type unbalance in order to obtain the relation between the
crosstalk in many transposition sections and that in one section.
In calculating this factor, attenuation cannot be neglected since long
lengths of line are involved.

* Two circuits are relatively transposed by one transposition at a given point in
the line. Transpositions in both circuits leave them relatively untransposed.

BELL SYSTEM TECHNICAL JOURNAL

The method of computing type unbalances for far-end crosstalk will
now be explained. As in the case of near-end crosstalk, the type
unbalance is defined by expressing the far-end crosstalk between two
long circuits as the product of the crosstalk coefficient, the frequency
in kilocycles and the type unbalance.

Figure 15B indicates the periodic variation with frequency of the
far-end crosstalk when type unbalance is controlling.

>: 40

(A)

/^s^

LINE ANGLE IN DEGREES (^D)
240 320 400 480 560

640

MEASURED FAR-END CROSSTALK

COMPUTED FAR-END CROSSTALK

(BJ

PAIRS 7-8,19-20
TRANSPOSED TO
TYPES H AND I

-'^

^_-,

PAIRS 13-14,33-34

TRANSPOSED TO^

TYPES LAND M

— ~,

N^,

-y

■^ """' '^ "

■r^

— -.

•,==,

— '

10 15 20 25 30 35

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 15 — Type unbalance and crosstalk vs. frequency and line angle in degrees.
For Part (5), see Fig. 27.4 for wire configuration and Fig. 28 for transposition
types.

In the case of near-end crosstalk, the method of computing the
type unbalance neglected interaction crosstalk since, ordinarily, the
transpositions needed to control transverse crosstalk make the inter-
action effect negligible. In the case of far-end crosstalk, the most
important type of interaction crosstalk is included in calculations of
type unbalances but another type of interaction crosstalk and the
direct transverse crosstalk are neglected. The transpositions needed
to properly suppress the important type of interaction crosstalk and
the indirect transverse crosstalk ordinarily make the neglected types
of crosstalk very small and the application of a more precise method
of computing type unbalances for far-end crosstalk is not justified in
practice.

OPEN-WIRE CROSSTALK

The far-end type unbalance for a non-transposed part of a long
parallel between two circuits will be computed first. Such a part of
a parallel is indicated by length D of Fig. 16. For purposes of compu-
tation this length is divided up into a number of short segments each
of length d. Considering the far-end crosstalk for two such segments
at the start of the length D it will be seen from the discussion of
crosstalk coefficients that transverse crosstalk in the length 2d will be

IFKd = 2{Fd + Fi)Kd.

In the above expression F is the far-end crosstalk coefficient, Fa being
that part due to direct crosstalk and Fi that part due to indirect
crosstalk.

TO LONG

circuits"

Fig. 16 — Far-end crosstalk between untransposed circuits in length D.

The above expression relates to the output-to-output crosstalk.
The input-to-output crosstalk is obtained by multiplying by the
propagation factor e"^^"' to allow for propagation from A to C. This
correction is usually made only when it is desired to obtain the input-
to-output crosstalk between complete circuits and it is usually satis-
factory to correct by using the attenuation factor and ignoring change
in phase.

The total transverse output-to-output crosstalk in the length D is:

{Fa + Fi)KD.
This is about equal to FiKD since Fd is ordinarily small compared to Fi.

52 BELL SYSTEM TECHNICAL JOURNAL

Figure 16 indicates with a solid line the important type of interaction
crosstalk between the first two segments by way of a representative
tertiary circuit c. As discussed in the section on crosstalk coefficients
and in Appendix A, the far-end crosstalk (output-to-output) of this
interaction type will be

NacNchKHn^)-'' = - 2yFiK<P approximately.

The interaction crosstalk as well as the transverse crosstalk is about
proportional to the indirect coefficient Fi.

Each segment of the disturbing circuit will have a similar interaction
crosstalk coupling with each preceding segment of the disturbed
circuit. The interaction crosstalk between segment EF and segment
BC \?> indicated on Fig. 16. The expression for this differs from the
above expression in that the additional propagation distance from E
to C and back must be allowed for. To get the total output-to-output
far-end crosstalk it is necessary to sum up all these interaction crosstalk
couplings between segments and to this sum add the total transverse
crosstalk in length D.

This clumsy summation process may be avoided by letting d be an
infinitesimal length and integrating between points A and G. This
results in the following approximate expression for the output-to-
output far-end crosstalk in the length D.

FjKD + FiKD + FiK

-2yD

- D

This assumes the same propagation constant for the disturbing,
disturbed and tertiary circuits. This approximation is justified for
short lengths of, say, 10 miles or less.

The last term represents the interaction crosstalk and this term is
negligible for small values of D. For larger values of D interaction
crosstalk must be considered and it is convenient to rewrite the
expression as follows:

1 _ e-270
FaKD + FiK ^

The first term representing the direct crosstalk is negligible for values
of D corresponding to a line angle of 90 degrees or less since Fd is
ordinarily small compared with Fi and D is not large compared with
(1 — e~2T^/27). Therefore, direct crosstalk ordinarily may be neg-
lected in computing far-end type unbalance. Another reason for
neglecting direct crosstalk is that it is readily cancelled by a few
relative transpositions while the remaining far-end crosstalk depends

OPEN^WIRE CROSSTALK

Upon the transpositions in a complicated way, because the various
interaction crosstalk couplings involve a variety of propagation
distances and, therefore, have a variety of phase angles. If both
circuits are transposed frequently but alike the direct crosstalk is not
affected by the transpositions but it is ordinarily small compared with
the indirect transverse crosstalk.

Figure 16 indicates by a dashed line another type of interaction
crosstalk involving the product of two far-end crosstalk couplings.
This effect can be neglected with practical arrangement of transpo-
sitions but may be important in the case of circuits having few trans-
positions or none at all.

In computing type unbalance the far-end crosstalk in an untrans-
posed segment of line of length D may, therefore, be written as:

FiK

2jD 1

— - FK-

--2 7©

approx.

Since the magnitude of Fi is ordinarily about equal to that of F, the
measured coefficient, it is usually satisfactory to use the latter value.

Fig. 17 — Far-end crosstalk in length 2D between circuits a and b with each circuit
transposed at the middle.

Having derived the above expression it is now possible to derive the
far-end type unbalance for two transposed circuits. Figure 17 indi-
cates a parallel between two long circuits. The type unbalance will
be computed for a length 2D in which both circuits are transposed at
the center. In the length 2D three far-end crosstalk paths must be
considered, that is, the far-end crosstalk in length AB, that in length

54 BELL SYSTEM TECHNICAL JOURNAL

BC and the important type of interaction crosstalk between length

BC and length AB. The output-to-output crosstalk values only will

be written for all these paths or, in other words, the effect of the

propagation distance A C will not be considered in the expressions.

The far-end crosstalk for either length AB or BC is given by the

above expression. Since both circuits are transposed at point B the

far-end crosstalk values in the two lengths will add directly and their

sum will be

1 _ e-'yD

2FK

Transmission from ^ to C through the crosstalk path in length AB
is reversed in sign due to the transposition in circuit b at B. The
output current of circuit a is also reversed in sign. In general, the
output-to-output current ratio may or may not be reve''sed in sign
depending on the transposition arrangement. It is convenient, how-
ever, to consider the first path as a reference and assign a plus sign to
the crosstalk. Other paths are then assigned the proper relative
phase angles.

As discussed in connection with Fig. 16, if the length D is very
short the interaction crosstalk between the two segments may be
written :

- 2y FiKD^ = - 2yFKD^ approx.

In practice the length D may be too long for this approximate ex-
pression in which case it is necessary to substitute for D in the above
expression the value derived in connection with the discussion of the
near-end crosstalk in a length D. In other words, D of the above
expression should be replaced by

1 - e-^y"

27 '

With this substitution the interaction crosstalk between the two
lengths becomes

(1 - 6-2^^)2

Transmission from A to C through this crosstalk path involves two
transpositions and therefore the sign of the above expression is not
reversed. Relative to the reference path through the crosstalk in
length AB the sign should be reversed, however, and become plus.
The total crosstalk in the length 2D is, therefore,

1 _ ^-2yD Cl _ -2yD\2 3 _ 4^-2yD I -47O

2FK^^^ + FK^ — ^ ^ = FK- ' ^ •

Z7 27 27

OPEN-WIRE CROSSTALK 55

The latter expression divided by F is the frequency times the far-end
type unbalance for the length 2D. If one of the circuits were trans-
posed at point B the crosstalk in length AB would be cancelled by
that in length BC. The sign of the interaction crosstalk between the
two lengths would be reversed and the expression would become

If neither circuit were transposed at B, the far-end crosstalk would
be that for a non-transposed length of 2D or:

1 _ ,-470

fk'—^ .

The frequency times the type unbalance values for the cases of one
transposition and no transpositions are the same (in magnitude) as
those derived for near-end crosstalk which were plotted (neglecting
attenuation) as curves 0 and P on Fig. 15A.

If both circuits are transposed at B the near-end type unbalance
remains the same as if there were no transpositions. The far-end type
unbalance is radically altered, however. This is evident if the above
equation is compared with that for the case of both circuits transposed.

This process of computing type unbalances may be extended from
two equal lengths to any number of equal lengths. It is necessary to
consider the interaction crosstalk between each length of the disturbing
circuit and each preceding length of the disturbed circuit. The rela-
tive propagation distances through the various interaction crosstalk
couplings must be taken account of.

Computations of far-end type unbalances are greatly simplified by
assuming the same propagation constants for the disturbing, disturbed
and tertiary circuits and by neglecting attenuation within a trans-
position section as in the case of near-end crosstalk. Since the tertiary
circuit may be composed of any combination of wires on the line or of
these wires and ground return, the propagation constant for a tertiary
circuit may be somewhat different from that for the disturbing and
disturbed circuits. This is particularly true of earth-return circuits,
but these are of little practical importance due to their relatively high
attenuation. All circuits not involving the earth have somewhere
near the same speed of propagation but the tertiary circuits may
differ greatly in attenuation constants.

For practical reasons a fair balance against crosstalk must be
obtained in each transposition section (eight miles or less) and, as in
the case of near-end crosstalk, type unbalances are calculated for the

56 BELL SYSTEM TECHNICAL JOURNAL

transposition arrangements which may exist in a single transposition
section. Since the attenuation in a transposition section is not great,
these calculations need not take into account differences in the attenu-
ation constants of the various tertiary circuits. A long line has a
series of transposition sections of various types and the total far-end
crosstalk for any two circuits is a summation of the crosstalk values
obtained from the type unbalances for the various sections plus
interaction crosstalk between the various combinations of sections.
With practical methods of transposition design, the transposition
arrangements are so chosen that the interaction crosstalk between two
sections is usually small compared with the far-end crosstalk in one
section. A long line for the most part consists of a succession of
similar sections with occasional sections of other types. Inter-
action crosstalk between dissimilar sections does not ordinarily
contribute appreciably to the total far-end crosstalk. For the im-
portant case of a succession of similar sections interaction crosstalk
between sections must be carefully considered since it may build up
systematically and the total may be large compared with the summa-
tion for the far-end crosstalk values for the individual sections.

Serious interaction crosstalk between similar sections is guarded
against by computing factors relating the far-end type unbalance in
one section to that in various numbers of successive sections with
various transposition arrangements at the junctions of sections. The
factors actually computed are somewhat in error since they involve
long distances and assume the same attenuation constants for all
circuits. The errors are not sufficient, however, to prevent the
factors from being a proper guide in avoiding systematic building up
of interaction crosstalk between sections.

The above discussion assumes that the tertiary circuits are indef-
initely extended or terminated to simulate their characteristic im-
pedance. The tertiary circuits may not be terminated at the ends of
a line since many of them are not used for transmission of speech or
signals. Complete reflections of the crosstalk current in the tertiary
circuits will, therefore, occur at their ends and these reflections some-
what modify the crosstalk currents in other circuits. This effect is
important in a very short line since the reflected wave is again reflected
at the distant end and at particular frequencies large changes in the
tertiary crosstalk currents may occur due to multiple reflections.
In a long line such multiple reflections are damped out and, in general,
tertiary circuit reflection effects are not important.

If all the pairs on a line are transposed for the same maximum
useful frequency, the transposed pairs will usually be relatively

OPEN-WIRE CROSSTALK 57

unimportant as tertiary circuits, that is, two pairs having small
crosstalk between them usually contribute but little to the crosstalk
between one of these pairs and any third pair. In some cases, however,
this effect is important. On some lines certain pairs may be transposed
for carrier operation and other circuits on the line for voice frequencies
only. A combination of the two kinds of circuits may have large
crosstalk between them at carrier frequencies and rnay contribute
appreciably to the carrier frequency crosstalk between the pair
transposed for carrier operation and some other pair also so transposed.
Far-end type unbalances which take account of transpositions in a
tertiary circuit must, therefore, be calculated. This can be done by
following the same general method discussed in connection with Fig. 17.
From the discussion of coefficients it follows that the far-end coefficient
for use in computing such a t^'pe unbalance will be:

^jf^E^ io-«,

where Nac and Neb are the near-end crosstalk coefficients for the
combination of disturbing circuit and tertiary circuit and the combi-
nation of tertiary circuit and disturbed circuit. Since these circuit
combinations involve recognized transmission circuits, their near-end
coefficients will be available since they must be measured or computed
in order to compute the near-end crosstalk.

If a parallel between two circuits is divided into a large number of
segments by transposition poles there is a wide variety of transposition
arrangements which may be installed at these poles. It is, therefore,
a complicated problem to devise charts and tables in reasonable
numbers which will cover all the possible type unbalance values for
the various transposition arrangements over a wide range of fre-
quencies. This is particularly true in the case of far-end type un-
balances since the type unbalance is altered by transposing both
circuits at the same points and it is necessary to work out a type
unbalance for each combination of transposition arrangements which
may be used in two circuits. In the case of near-end crosstalk a
number of different transposition arrangements will have the same
type unbalance since only the relative transpositions need be con-
sidered.

The circuit capacity of a line may be increased by the use of phantom
circuits (generally when carrier-frequency systems are not involved)
which must, of course, be transposed to avoid noise and crosstalk.
The crosstalk between phantom circuits may be calculated in a manner
similar to that for pairs. The calculation of crosstalk between side

58 BELL SYSTEM TECHNICAL JOURNAL

circuits of the phantoms or between a side circuit and a phantom
circuit is complicated by the fact that the phantom transpositions
cause the side circuits to change pin positions. Near-end and far-end
type unbalances have been computed, however, which take account
of this "pin shift" effect of the phantom circuits. In general, the
use of phantom circuits seriously limits the crosstalk reduction which
may be obtained by transpositions. Phantom circuits are often
uneconomic since they seriously restrict the number of carrier fre-
quency channels which may be operated over a given pole line.

As indicated by Fig. 15 A the values of type unbalance times fre-
quency have marked maximum and minimum values when they are
plotted against frequency or length. The maximum values are usually
reduced by increasing the number of transpositions in a given length.
When there are a number of circuits on the line it is usually necessary
that the propagation of current between successive transposition poles
does not change the phase by more than about five degrees. Since
the phase change is about two degrees per mile per kilocycle the
maximum transposition interval in miles is about 2.5//^ where Fis the
frequency in kilocycles. This means .25 mile or 1300 feet at 10
kilocycles and .06 mile or 300 feet at 40 kilocycles.

It does not follow, however, that the least maximum value of type
unbalance for a range of frequencies is obtained by using the greatest
number of transpositions for a given number of transposition poles.
This is illustrated by Fig. 15A which shows that the least maximum
value is obtained with no transpositions rather than with one trans-
position. The total crosstalk current at a terminal is composed of
numerous elements of various magnitudes and phase relations. The
vector sum of these elements tends to be small at particular frequencies
with no transpositions at all and it is important to preserve this
tendency as much as possible when choosing an arrangement of
transpositions. The vector sum of the elements can never be made
zero since this would require that the circuits have no attenuation
and infinite speed of propagation. This sum and, therefore, the type
unbalances can be made very small, however, by choosing a suitable
transposition arrangement and making the interval between trans-
position poles very small. In practice, the values of type unbalance
times frequency for adjacent circuits are restricted to values much
less than those of Fig. 15A.

Vacuum Tube Electronics at Ultra-high Frequencies *

By F. B. LLEWELLYN

Vacuum tube electronics are analyzed when the time of flight of the
electrons is taken into account. The analysis starts with a known current,
which in general consists of direct-current value plus a number of alter-
nating-current components. The velocities of the electrons are associated
with corresponding current components, and from these velocities the
potential differences are computed, so that the final result may be expressed
in the form of an impedance.

Applications of the general analysis are made to diodes, triodes with
negative grid, and to triodes with positive grid and either negative or posi-
tive plate which constitute the Barkhausen type of ultra-high-frequency
oscillator. A wave-length range extending from infinity down to only a
few centimeters is considered, and it is shown that even in the low-frequency
range certain slight modifications should be made in our usual analysis of
the negative grid triode.

Oscillation conditions for positive grid triodes are indicated, and a brief
discussion of the general assumptions made in the theory is appended.

I. Foreword

THE art of producing, detecting, and modulating ultra-high-fre-
quency electric oscillations has reached the same state of develop-
ment which was attained in early work on lower frequency oscillations
when experiment had outstripped theory. The experimenters were
able to produce oscillations by using vacuum tubes, but were not
able to explain why. They were able to make improvements by the
long and tedious process of cut and try, but did not have the powerful
tools of theoretical analysis at their command. In particular, the
advantage of the theoretical attack may be illustrated by the rapid
advance in technique which followed the theoretical concept of the
internal cathode-plate impedance of three-element vacuum tubes.
The work of van der Bijl and Nichols showed that for purposes of
circuit analysis this path could be replaced by a fictitious generator
of voltage, fiCg, having an internal impedance whose magnitude is
given by the reciprocal of the slope of the static Vp — Ip characteristic.
Development of commercially reliable vacuum tube circuits began
forthwith. In a similar, yet less complicated manner, the internal
network of two-element tubes may be replaced by an equivalent
resistance when relatively low frequencies only are considered.

In these concepts where the vacuum tube is replaced by its equiva-

* Presented in brief summary before U. R. S. I., Washington, D. C, April, 1932.
Proc. I. R. E., Vol. 21, No. 11, November, 1933.

60 BELL SYSTEM TECHNICAL JOURNAL

lent network impedance, one outstanding feature is exemplified:
namely, the separation of the alternating- and direct-current com-
ponents. The equivalent networks are applicable to the alternating-
current fundamental component of the current and differ widely from
the direct-current characteristics. A complete realization of the im-
portance of this separation will be of advantage in the later steps
where extension of the classical theory to the case of ultra-high-
frequency currents is described.

For a short time after the original introduction of the equivalent
network of the tube, affairs progressed smoothly. Soon, however,
frequencies were increased and a new complication arose. The diffi-
culty was attributable to the interelectrode capacities existing between
the various elements of the vacuum tube. The original attempts to
take this into account were based on the viewpoint that the tube
network should be complete in itself and separate from the external
circuit network to which it was attached. Correct results, of course,
were obtained by this method but later developments showed the
advantage of considering the equivalent network of the complete
circuit, including both tube and external impedances in a single net-
work. For instance, by grouping the combination of grid-cathode
capacity with whatever external impedance was connected between
these two electrodes, a great simplification occurred. This step also
has its analogy in the development of ultra-high-frequency relations.

As time went on, higher and higher frequencies were desired, and
they were produced by the same kind of vacuum tubes operating in
the same kind of circuits, although refinements in circuit and tube
design allowed the technique to be improved to the point where
oscillations of the order of 70 to 80 megacycles were obtainable with
fair efficiency. When the frequency was increased still further, it was
found that extension of the same kind of refinements was unavailing
in maintaining the efficiency and mode of operation of the higher
frequency oscillations at the level which had previously been secured.
Ultimately, the three-electrode tube regenerative oscillator ceases to
function as a power generator in the neighborhood of 100 megacycles
for the more usual types of transmitting tubes. When this point was
reached, the external circuit had not yet shrunk up to zero proportions
and neither had its losses become sufficiently high to account altogether
for the failure of the tube to produce oscillations. From this point on,
the old-time cut-and-try methods were employed and marked im-
provements were secured. In fact, low power tubes have been made
which operate at wave-lengths of the order of 50 to 100 centimeters
with fair stability, although quite low efficiency.

VACUUM TUBE ELECTRONICS 61

In the meantime, the production of ultra-high-frequency oscilla-
tions had been progressing in a somewhat different direction. The
discovery, about 1920, by Barkhausen that oscillations of less than
100 centimeters wave-length could be secured in a tube having a
symmetrical structure, when the grid was operated at a fairly high
positive potential, while the plate was approximately at the cathode
potential, started experiments on what was thought to be an altogether
different mode of oscillation. Workers by the score have extended
both the experimental technique and the theory of production of this
newer type of oscillation. However, one of the results which an
analysis of ultra-high-frequency electronics illustrates is that the elec-
tron type of oscillator is merely another example of the same kind of
oscillation which was produced in the old-time so-called regenerative
circuits.

For the purpose of extending the theory of electronics within vac-
uum tubes to frequencies where the time of transit of the electrons
becomes comparable with the oscillation period, it is important at the
outset to select an idealized picture which is simple enough to allow
exact mathematical relations to be written. At the same time, the
picture must be capable of adaptation to practical circuits without
undue violence to the mathematics. An example of this kind of
adaptation is illustrated by the classical calculation of the amplification
factor fx, which was accomplished by consideration of the force of the
electrostatic field existing near the cathode in the absence of space
charge even though tubes were never operated under this condition.
In a like manner, such violations of the ideal must, of necessity, be
made in ultra-high-frequency analysis but their practical validity lies
in so choosing them that the quantitative error introduced is less than
the expected precision of measurement. It becomes, therefore, of the
utmost importance to state clearly the transitions which occur be-
tween results obtained for the idealized case to which the mathematics
is strictly applicable and the practical circuits where the assumptions
and approximations are made to conform with operating conditions.

A start has already been made on the problem of developing such
a generally valid system of electronics. This was done by Benham ^
who considers a special case comprising two parallel-plane electrodes,
one of which is an emitter and the other a collector, when conditions
at the emitter are restricted by the assumption that the electrons are
emitted with neither initial velocity nor acceleration. This work of
Benham's has the utmost importance in a general electronic theory

^ W. E. Benham, "Theory of the Internal Action of Thermionic Systems at
Moderately High Frequencies," Part I, Phil. Mag., p. 641; March (1928); Part II,
Phil. Mag., Vol. 11, p. 457; February (1931).

62 BELL SYSTEM TECHNICAL JOURNAL

and, in fact, the means of extending his theory exists primarily in the
selection of much more general boundary conditions than were as-
sumed by him. It will, therefore, result that some repetition of
Benham's work will appear in the following pages. However, in view
of the new state of the theory and the importance of accurate founda-
tions for it, this repetition is advantageous rather than otherwise.

With these preliminary remarks in mind, the next step is the selec-
tion of the idealized starting point for a mathematical analysis. Ex-
actly as was done by Benham we take two parallel planes of infinite
extent, one of which is held at a positive potential V with respect to
the other, and between the two electrons are free to move under the
influence of the existing fields. The next step in the idealization con-
stitutes the separation of alternating- and direct-current components
not only of current and potential, but also of electron velocity, charge
density, and electric intensity. With this separation, the restriction
that the direct-current component of the electron velocity and acceler-
ation is zero at the negative plane may be made while leaving us free
to select much more general boundary conditions for the alternating-
current component. It is true that the more general conditions now
proposed will not fit the original physical picture where the negative
plane consists of a thermionic emitter. Nevertheless the extension is
of importance since it allows application to be made to the wide
number of physical cases where "virtual cathodes" are formed. One
such example is the convergence of electrons toward a plate maintained
at cathode potential while a grid operating at a high positive potential
with respect to both is interposed between them. In a stricter mathe-
matical sense, the broader boundary conditions come about because
of the fact that the general equations containing all components are
separable into a system of equations, one for each component, and
that the boundary conditions for the different equations of the system
are independent of each other.

The concept of an alternating-current velocity component requires
a few words of explanation. In the absence of all alternating-current
components, electrons leave the cathode with zero velocity and acceler-
ation and move across to the anode with constantly increasing velocity
under the well-known classical laws. This velocity constitutes the
direct-current velocity component. When the alternating-current
components are introduced, there will be a fluctuation in velocity
superposed on the direct-current value, and the alternating-current
component need not be zero at a virtual cathode. This separation of
components will come about naturally in the course of the mathe-
matical analysis which follows, but since the interpretation of the

E =

dV
dx

)

47rP,

J =

PU +

dE
dt '

VACUUM TUBE ELECTRONICS 63

equations is of paramount importance, a few words of explanation and
repetition will be necessary,

II. Fundamental Relations

For the development of the fundamental relations existing between
the two parallel planes, we have the classical equations of the electro-
magnetic theory which may be set down in the following form;

(1)

where E is the electric intensity, V the potential, P the charge density,
/ the total current density consisting of conduction and displacement
components, and U is the charge velocity. These equations apply to
frequencies such that the time which would be taken by an electro-
magnetic wave in traveling between the two planes is inappreciable
when compared with the period of any alternating-current frequency
considered. Ordinarily this limitation will become of importance only
at frequencies higher even than those in the centimeter wave-length
range where the time of electron transit is of great importance, al-
though the time of passage of an electromagnetic wave is still negligibly
small.

An electron situated between the two parallel plates will be acted
upon by a force which determines its acceleration. The resulting
velocity is a function both of the distance, x, from the cathode and
the time, /, so that in terms of partial derivatives, the equation
expressing the relation between the force and acceleration is

f+C/|^=^£. (2)

dt dx m

From (1) and (2) may readily be obtained

17'7"\t7" at / 't\

di dx j m '

In this equation we have a relation between the velocity and the total
current density. The advantage of this form of equation for a starting
point lies in the fact that the total current density / is not a function

64 BELL SYSTEM TECHNICAL JOURNAL

of X. This comes about because of the plane shape and parallel dis-
position of the electrodes, and the fact that current always flows in
closed paths. Thus, while the current between the two planes may
be a function of time, it is not a function of x.

The separation of alternating- and direct-current components may
now be made. We write

/ = /o + /l + /2 + • • • (4)

with corresponding

^ = Z7o + t/i + Z7, + • • • , I
V = Vo + Fi + F2 + • • • , I

where the quantities with the zero subscript are dependent on x, only,
those with subscript 1 are dependent to first order of small quantities upon
time, those with subscript 2 are dependent to second order, and so forth.
As a result of this separation in accord with the order of dependents upon
time, (3) may be split up into a system of equations, the first of which
expresses the relation between Uo, Jo, and x and does not Involve time.
This is the relation governing the direct-current components. The
second equation of the system involves the relation between Ui, Ji, x,
and time, and contains Z7o which was determined by the first equation.
Likewise, the third equation contains U2, U\, J2, x, and t. Since the
series given by (4) and (5) are convergent so that, in general, the terms
with higher order subscripts are smaller than those with lower sub-
scripts, we may consider that, at least for small values of alternating-
current components, the total fundamental frequency component is
given by the terms with unity subscript.

The first two equations of the system are as follows:

l/.f(c/„^) = 4.i/., (6)

dx \ OX J m

d , jj d \/dUi . ,, dUi. ., dUo

+ <.(^o^^«) = 4.^... (7)

In the solution of (6), the boundary conditions are restricted so
that when x is zero, the velocity and acceleration both are zero. These
restrictions mean that initial velocities are neglected, and that com-
plete space charge is assumed. Thus the solution for Uo is

Uo = ax"\ (8)

VACUUM TUBE ELECTRONICS 65

where

„ = (l8,^ /.)'"• (9)

The solution of (7) is more complicated. We assign a particular
value to /i, namely, Ji = A sin pt and find the corresponding value of
Ui. To do this, it is convenient to change the variable x to a new
variable ^, which will be called the transit angle. This new variable
is equal to the product of the angular frequency p and the time r
which it would take an electron moving with velocity Uo to reach the
point X and is given as follows:

^ = ^r=^xi/3. (10)

Upon changing the dependent variable from Ui to w, where Ui = co/^,
we find from (7)

-^ + i>^)'^ = ^/3sin^/, (11)

where

This has the solution

^ = 47r- A.
m

f/i =

sin pt + 7 cos pt + 7^,(^ - pt) + 7 i^2(^ - pt)

(12)

This equation contains two arbitrary functions of (^ — pt) which must
be evaluated by the boundary conditions selected for Ui. Thus the
boundary conditions for the alternating-current component make their
first appearance.

From the form of (7) which is linear in Ui, it is evident that Ui
must be a sinusoidal function of time having an angular frequency p
in order to correspond with the form of /i. It follows, then, that the
most general form which can be assumed for the steady state functions
Fi and F^ is as follows:

Fi(^ - pt) = a sin (^ - pt) -{-b cos (^ - pt)\ .^^.

Fi{^ - pt) = c sin (^ - pt) + d cos (^ - ^/) j

Now for the boundary conditions. As pointed out, there is no
mathematical necessity for the boundary conditions imposed upon Ui
to correspond with those which were imposed upon Uq. At an actual
cathode consisting of an electron emitting surface it would be appro-
priate to assume that the initial velocities are in no way dependent
upon the current, but we shall have to deal not only with actual

66 BELL SYSTEM TECHNICAL JOURNAL

cathodes, but also with virtual - cathodes where the assumption of zero
alternating-current velocity and acceleration is unwarranted. Such a
virtual cathode might occur, for instance, between a grid operated at
a positive direct-current potential and a plate nearly at cathode
potential. If enough electrons came through the mesh of the grid
to depress the potential until it became practically zero at some point
in the space between grid and plate, the direct-current boundary con-
ditions of zero velocity and acceleration of electrons would be fulfilled
at that point. The general equations for the alternating current will
therefore apply when the origin is taken at the point of direct-current
potential minimum which forms the virtual cathode, and when all of
the electrons which are emitted by the actual cathode pass by the
virtual cathode and reach the plate. In the event that some of the
electrons are turned back at the virtual cathode and move again
toward the grid, as indeed they all do when the plate is at a negative
potential, a change in the form of the general equation is necessary,
and will be described in the sections dealing particularly with positive
grid triodes. This change, however, affects merely the form of the
equations and not the physical arguments underlying the selection of
boundary conditions, which are the same whether all the electrons
reach the plate or whether some or all of them turn back toward the
grid.

If the alternating-current velocity is determined by small varia-
tions in grid potential, let us say, it is evident that no additional
assumptions save the requirement that the velocity must not become
infinite may be made concerning its value at the virtual cathode.
Consequently, a quite general set of boundary conditions will suffice
to determine the quantities, a, b, c, d, which appear in (13) and thus
completely determine Ui.

Since there are two arbitrary functions in (12), two boundary con-
ditions will be needed. Further inspection shows that the stipulation
that the alternating-current velocity be finite at the origin is sufficient
to furnish one of these boundary conditions. For the other, a knowl-
edge of the value of the alternating-current velocity at any point
between the two reference planes is sufficient. Thus, if at a particular
value of ^, say ^i, we know that Ui is equal to M sin pt -\- N cos pt,
we have enough information to calculate its value at all other points
between the two planes. For example, the two reference planes might
be the grid and plate of a positive grid triode. In this event, the
alternating-current velocity at the grid could be calculated at the
grid plane by means of conditions between there and the cathode.

= E. W. B. Gill, "A Space-Charge Effect," Phil. Mag., Vol. 49, p. 993 (1925).

VACUUM TUBE ELECTRONICS 67

In mathematical form the two boundary conditions may be set
forth as follows:
when,

^ = 0, Ui must be finite, (14)

^ == li, Ui = Msmpt + N cos pt. (15)

From (12) and (13) these result in the values:

c = 0, d = - 2,

a = |- (M cos ^1 - A^ sin ^i) + cos ^1-71 sin ?i, (16)

6 = I (1 - cos ^1) - sin ^1 - ^ (Msin ^1 + TV cos ^1). (17)

Thence from (12) we have for the alternating-current velocity, in
general ,

Ui = {M + iN) (cos ^1 + i sin ^1) (cos | - i sin ^)

+ ^2 {(cos ^1 - |sin lij - i (I - |cos ^1 - sin ?ij|

(cos ^ - ? sin ^) - (1 - |sin A - i- (1 - cos ^)] , (18)

where, in accord with engineering practice, complex notation is em-
ployed, so that sin pt has been replaced by e'^' and cos pt has been re-
placed by ie^p\ where i = V— 1.

The first step in the derivation of fundamental relations has now
been achieved. The alternating-current velocity at any point between
the two planes has been expressed in terms of the alternating-current
velocity, M -f iN, existing at a definite value of x, say Xi, correspond-
ing to the transit angle ^1.

The next step is a determination of the potentials corresponding
to the velocities Uo and Z7i, respectively. Thus from (1) and (2)

_£^=^+y«' (19)

m dx at ox

and then with the separation of components as given by (5)

68 BELL SYSTEM TECHNICAL JOURNAL

The solution of (20) is

Fo= -^'W^ -^^V/^ (22)

2e Ze

which is the well-known classical relation between the potential, the
current, and the position between two parallel planes where complete
space charge exists. The complete space-charge condition is postu-
lated by the boundary conditions selected for Uq and the implications
involved are discussed by I. Langmuir and Karl T. Compton.^

The alternating-current component of the potential is obtained by
integration of (21) as follows:

- ^ Fi = I- {u,dx + U,U, +m, (23)

m ot J

whence, from (18), and in complex notation

y^= - — ?^2 (^ + ^^)(cos ^1 + i sin ^i)[(^ sin ^ -f cos k)
e yp^

-f ^T^cos ^ - sin I)]

2ma^^\\( ^ 2..\ ./2 2 . ..\1

[(^ sin ^ + cos ^) + i{^ cos ^ - sin ^)]

- cos ^ - i{^ + W - sin ^)

+ constant. (24)

With the attainment of (24), the fundamental relation between the
alternating-current component /i and the alternating-current poten-
tial Vi in the idealized parallel plate diode has been secured. In a
more general sense the equation is applicable between any two fictitious
parallel planes where one is located at an origin where the boundary
conditions for Uq are satisfied; namely, that the direct-current com-
ponents of the velocity and acceleration are zero, and the value of the
alternating-current velocity at a point, .ri, corresponding to the transit
angle, ^i, is given by M sin pt -\- N cos pt, or by M -f iN in complex
notation.

Equation (24) contains an additive constant which always appears
in potential calculations. This constant disappears when the potential
difference is computed. For instance, suppose the potential difference
between planes where ^ has the values ^ and ^', respectively, is desired.

" I. Langmuir and Karl T. Compton, "Electrical Discharges in Gases" — Part II,
Rev. Mod. Phys., Vol. 3, p. 191; April (1931).

VACUUM TUBE ELECTRONICS 69

We have

Vi = /(?) + constant,

Vi = M) + constant,
so that

Vx- F/ ^M) -fin. (24-a)

Since the potential difference is always required rather than the
absolute potential, (24-a) gives the means for applying (24) to actual
problems.

III. Application to Diodes

In the application of the fundamental relations to diodes where the
thermionic emitter forms the plane located at the origin and the anode
coincides with the other plane, the boundary condition is that Ui shall
be zero at the cathode. This means that both M and N are zero and
that ^1 is also zero. The resulting forms taken by (18) and (24-a),
respectively, are as follows:

^1= S2

V, - F/
~ e9p*

2 . \ . / 2 . 2

1 + cos ^ — --sin ^ j -\- i i -r — sin ^ — 7 cos ^

(25)

C(2cos^ + ^sin^-2)+^(?+|?^-2sin?-|-^cos?)]. (26)

These two equations are identical with those obtained by Benham,^
and graphs are given in Figs. 1 and 2 showing their variation as a
function of the transit angle ^. In particular, the equivalent impe-
dance between unit areas of the two parallel planes may be found from
(26). It must be remembered that the current. A, was assumed
positive when directed away from the origin. Hence, we may write

Z=-Zl^. (27)

Moreover, the coefficient outside the square brackets in the equation
may be expressed more simply when it is realized that the low-fre-
quency internal resistance of a diode is given by the expression

^0 = "~ ^T" ' (28)

the minus sign again appearing because of the assumed current direc-
tion. Consequently, under the condition of complete space charge,

BELL SYSTEM TECHNICAL JOURNAL

we have from (22)

2ma^^ 12roA

e9p^

(29)

In addition to the graphs in Figs. 1 and 2 showing the real and
imaginary components of impedance and velocity, the graphs shown
in Figs. 3 and 4 give their respective magnitudes and phase angles.

■0.40
• 0.60
•0.80
- 1.00

~~~"

Zp = Pp + ixp

r-o

Vv^

■ xp
"■o

F"ig. 1 — Plate impedance of diodes or of negative grid triodes as a
function of electron transit angle.

The impedance charts show a negative resistance for diodes in the
neighborhood of a transit angle, ^, of 7 radians. The possibility of
securing oscillations in this region has been discussed by Benham, so
that only a few additional remarks will be made here.

The magnitude of the ratio of reactance to resistance is about 15
when the transit angle is 7 radians. This means that oscillation con-
ditions require an external circuit having a larger ratio of reactance to
resistance. On account of the high value of reactance required, a
tuned circuit or Lecher- wire system is needed, which would have to
operate near an antiresonance point in order to supply the high reac-
tance value. But the resistance component of the external circuit
impedance is large at frequencies in the neighborhood of the tuning
point, so that the ratio of reactance to resistance is small. Calcula-

VACUUM TUBE ELECTRONICS

tions show that the possibility of securing external circuits having low
enough losses to meet the oscillation requirements of most of the
diodes which are at present available is not very favorable. The large
radio-frequency loss in the filamentary cathodes with which many
tubes are supplied is an additional obstacle to be overcome before
satisfactory ultra-high-frequency operation of diodes can be expected.

1.0

0.9

-0.1

-02

-03

-04

-Q5

-06

-0.7

-OS

-09

-1.0

u, =

(0 +

9J0

:ii*

8 9 10 11 12 13

Fig. 2 — Electron velocity fluctuation in diodes versus transit angle.

IV. Triodes with Negative Grid and Positive Plate

In the application of the fundamental relations to triodes operating
with the grid at a negative potential, the problem becomes more com-
plicated because of the several current paths which exist within the
tube. Moreover, the direct-current potential distribution is disturbed
in a radical way by the presence of the negative grid. In fact, the

BELL SYSTEM TECHNICAL JOURNAL

negative grid triode in some respects offers greater theoretical difficulty
than does the positive grid triode, which is treated in the next section.
However, because of the greater ease in the interpretation of the re-
sults in terms which have become familiar through years of use, the
negative grid triode is treated first.

too

I To I

0.20

0.10

Zp = |Zp|eiep

"~-

-5 <

Fig. 3 — Magnitude and phase angle of plate impedance of diodes or of
negative grid triodes versus transit angle.

In the analysis recourse must be had to approximations and ideali-
zations which allow the theory to fit the practical conditions. In the
selection of these, the first thing to notice is that no electrons reach
the grid, so that most of the electrostatic force from the grid acts on
electrons quite near the cathode, where the charge density is very great.
The most prominent effect of a change in grid potential will thus be a
change in the velocity of electrons at a point quite near the cathode.
It will thus be appropriate to assume as a starting point that the
alternating-current velocity at a point Xi, located quite near the
cathode is directly proportional to the alternating-current grid poten-
tial, Vg, so that we may write,

when

£/i = (M + iN) = k Vg.

(30)

VACUUM TUBE ELECTRONICS

In any event, this relation may be justified if the factor of proportion-
aHty, k, be allowed to assume complex values, and ^i is not taken too
near the origin. Actually, the electron-free space surrounding the grid
wires, and the fact that the electric intensity at a point midway be-
tween any two of the wires is directed perpendicularly to the plane of
the grid, gives us more confidence in extending the approximation, so
that k will be regarded as real, and ^i will be taken very small.

1.0
0.9
Q8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

i = y

a 2 + K 2

aUo , r.

/'ya2+b2

6 7

10 11 12 13

-2
-3
-4
-5
-6
-7

Fig. 4 — Magnitude and phase angle of electron velocity fluctuation
in diodes versus transit angle.

Equation (24) may, therefore, be applied under the conditions that
^1 -^ 0, and gives the following for the potential diliference between
plate and cathode:

llrpA

(^ sin ^+2 cos ^-2)+f(^ + i^3_2 sin ^-f^cos ^

- (M+iW) ^ [(t sin ^+cos ^-\)-i{sm ^-^cos^)]

(31)

74 BELL SYSTEM TECHNICAL JOURNAL

This equation may be written in condensed form with the aid of (30)

where

Fp = Jx{r + ix) - F,(m + iv),
r = -^asin^ + 2cos^-2),
x= -^{^ + W - 2sin^ + ^cos^),

(32)

M = -z^ (^ sin ^ + cos ^ - 1),

2^40

2jUo

J' = — ^ (?cos ^ — sin ^).

{?>?>)

1.00
0.80

/M-o

a = |x+ iv

0.20

-0.20

M-o

Fig. 5 — Real and imaginary components of complex amplification factor
of negative grid triodes versus transit angle.

The significance of (32) is at once apparent when it is compared
with the classical form of the equation representing the alternating-
current plate voltage, namely,

F„

/„;-

?>'0

y.Vo.

The plate resistance ro has now become complex as likewise has the
amplification factor ix. Values of the plate impedance

Zp = r + ix

VACUUM TUBE ELECTRONICS

are the same as those obtained for the diode and are plotted in Figs. 1
and 3. Values of the ampHfication factor

0" = ^ + iu

are shown in Figs. 5 and 6.

-1

-2
-3

-5

0--

\o\e

i<j,

-7
-8

-10
-11

Fig. 6— Phase angle of amplification factor of negative
grid triodes versus transit angle.

It is evident that radical changes in the phase angles existing be-
tween the grid voltage and plate current are present when the transit
time becomes appreciable in comparison with the period of the applied
electromotive force. The plate impedance decreases in magnitude as
also does the magnitude of the amplification factor. However, the
ratio of the two, namely, a/z, maintains a fairly constant magnitude
as shown in Fig. 7, whose phase angle nevertheless rotates continually
m a negative direction becoming equal to 3 radians when t is 27r.

The interelectrode capacity between cathode and plate is included
m the fundamental relations here employed. This inclusion exhibits
one important difference between (32) and the classical case. At low
frequencies, the equivalent circuit represented by (32) degenerates into
that shown on Fig. 8. The capacity branch exists in parallel with the

BELL SYSTEM TECHNICAL JOURNAL

resistive branch and they are both in series with the effective generator
o-gp, whereas in the classical picture the capacity branch shunts the
effective generator and plate resistance which are in series with each
other. Practically the difference between the two equivalent circuits
is negligible except at extremely high frequencies. The following
physical viewpoint supports the newer picture.

Fig. 7 — Magnitude of complex mutual conductance of negative
grid triodes versus transit angle.

As pointed out, the action of the grid is exerted mostly on the region
of dense space charge existing very near the cathode and variations
in the grid potential act on the velocities of the emerging electrons,
thus producing the equivalent generator of the plate circuit. The
plate current consists of conduction and displacement components
whose sum is the same at all points in the cathode-plate path. Near
the cathode, the conduction component comprises the whole current
because of the high charge density and the effective generator acts in
series with this current and hence in series with the path of the dis-
placement current into which the character of the total current gradu-
ally changes as the plate is approached.

Strictly speaking, the equivalent circuit corresponding to (32) ex-
ists, not between the plate and cathode, but between the plate and
the potential minimum near the cathode which is caused by the finite

VACUUM TUBE ELECTRONICS

velocities with which electrons are emitted from the cathode. Prac-
tically, the difference is negligible except at extremely high frequencies.
Since the impedance between the cathode and potential minimum is
small compared to the plate impedance, its effect is merely to add a
loss to the system which increases with frequency since the plate im-
pedance approaches a capacity as the frequency approaches infinity.
The grid cathode path presents less difficulty, although a somewhat
less rigorous treatment is given here. As pointed out, the force from
the grid acts on the high charge density region existing near the
potential minimum. The impedance between cathode and grid, there-
fore, consists of two parts in series; namely, capacity between grid and
potential minimum and impedance between potential minimum and
cathode, the latter part of this impedance being common both to plate-
and grid-current paths.

Zd = To + iXr

aeg = (ix+iv)eg

(M,+-iv)i

4
C=-yC,

l-=-[^C,ro2
M'=M'0

■o

Fig. 8 — Equivalent network of plate-cathode path of negative grid
triodes for transit angles less than 0.3 radian.

If we were to connect the grid and cathode terminals of such a
triode to a capacity bridge and measure the capacity existing there
when the tube was cold and when the cathode was heated, we should
find that the capacity would exhibit a slight increase in the latter case.
The reason for this increase may best be explained by noting that in
the cold condition the electrostatic force from the grid is exerted on the
cathode itself, whereas in the heated state, the force acts on the elec-
trons near the potential minimum, thus resulting in an increased capac-
ity in series with a resistive component.

In some measurements of the losses in coils which were made at a
frequency of 18 megacycles, J. G. Chaffee of the Bell Telephone Lab-
oratories has found that a loss existed between grid and cathode of
vacuum tubes which was much greater than can be accounted for by
any of the dielectrics used and which was present only when the tube

78 BELL SYSTEM TECHNICAL JOURNAL

filament was hot. This loss increased with frequency in the manner
characteristic of that of the capacity-resistance combination between
cathode and grid which was described above. Present indications are
that, at least in part, the loss may be ascribed to the resistance existing
between the cathode and the region of potential minimum.

Of the three current paths through the tube, one more still remains
to be considered. This is the grid-plate path. The relations involved
here are more readily seen by considering first a low-frequency ex-
ample. Here the electron stream passes through the spaces between
grid wires, afterward diverging as the plate is approached. Electro-
static force from the grid acts not only on the plate but also on the
electrons in the space between. It is evident, then, that the path
which, when the cathode was cold, constituted a pure capacity changes
into an effective capacity different from the original in combination
with a resistive component. The losses would be expected to increase
with frequency just as they did in the grid-cathode type. The change
in grid-plate impedance is particularly noticeable when it is attempted
to adjust balanced or neutralized amplifier circuits with the filament
cold, in which case the balance is disturbed when the cathode is heated.

As yet, no accurate expression for this grid-plate impedance has
been obtained, either at the low frequencies where transit times are
negligible or at the higher frequencies now particularly under investi-
gation. The reason for this lies in the repelling force on the electron
stream of the negative grid so that the assumption of current flow in
straight parallel lines is not valid in so far as current from the grid to
the plate is involved.

It has been shown that both the cathode-grid path and the grid-
plate path contain resistive components with corresponding losses
which increase with increase of frequency. This loss may be cited as
a reason why triodes with negative grids cease to oscillate at the higher
frequencies. If it were not for these losses, external circuits could be
attached to the tube having such phase relations as to satisfy oscilla-
tion conditions, so that the negative grid triode could be utilized in
the range which is now covered by the triode with positive grid.

V. Triodes with Positive Grid and Slightly Positive Plate

When the grid of a three-element tube is operated at a high positive
potential with respect both to cathode and plate, electrons are at-
tracted toward the grid, and the majority of them are captured on their
first transit. Those which pass through the mesh and journey toward
the plate will be captured by the plate if its potential is sufficiently
positive with respect to the cathode.

VACUUM TUBE ELECTRONICS

In general, space-charge conditions existing between grid and plate
are quite complicated. An analysis has been made by Tonks * which
indicates several distinct classes of space-charge distribution which are
possible. In the first place so few electrons may pass the grid mesh
that no appreciable space charge is set up between there and the plate.
In this instance a positive plate will trap them all, whereas a negative
plate will return them all toward the grid. Second, with a fixed posi-
tive plate potential an increase in the number of electrons which pass
the grid mesh will result in a depression of the potential distribution
as illustrated at (a) by the curves in Fig. 9. This depression will con-

Fig. 9 — Potential distributions in positive grid triodes.

tinue to increase until a potential minimum is formed. When this
potential minimum becomes nearly the same as that of the cathode,
either of several things may occur. If the minimum is just above the
cathode potential, all electrons will pass that point and eventually
reach the plate. However, an extremely small increase in the number
of electrons will cause the potential minimum to become equal to the
cathode potential. When this happens some of the electrons will be
turned back and travel again toward the grid. These will increase the
charge density existing and, therefore, cause a further depression in
the potential resulting in a mathematical discontinuity so that the

^ L. Tonks, "Space Charge as a Cause of Negative Resistance in a Triode and
Its Bearing on Short-Wave Generation," Pliys. Rev., Vol. 30, p. 501; October (1927).

80 BELL SYSTEM TECHNICAL JOURNAL

curve of the potential suddenly changes its shape with a resulting
change in plate current. Again, the plate may be operated at a
negative potential. In this case, none of the electrons will reach it
and the potential distribution curves have the character illustrated
at [c) and (d) in Fig. 9.

In attempting to apply the fundamental relations to this grid-plate
region, we must choose our origin at a point where the potential dis-
tribution curve touches the zero axis and is tangent to it. Whenever
such a point exists, the relations may be applied as described below.
Even when this condition does not exist inside the vacuum tube, there
may be a virtual cathode existing outside of the plate.

Whenever all of the electrons passing the grid reach the plate the
general equations may be applied in a straightforward manner with
the origin taken at the virtual cathode. Whenever some of the elec-
trons are turned back toward the grid, slightly different equations are
required, although they may be applied in the same manner. These
modified equations will be derived and discussed after the application
of the equations already derived has been made to the case where all
of the electrons reach the plate.

Choosing the origin for this latter case at the point of zero potential
or virtual cathode, we can compute the impedance between the grid-
plane and the virtual cathode when we know the alternating-current
velocities with which the electrons pass through the grid-plane. This
has been found for the condition of complete space charge between
cathode and grid and was given by (25). Likewise, it can be found on
the supposition that no space charge exists in the cathode-grid region
and the result will be calculated later. Thus, two limiting cases are
available for numerical application.

In order to prevent confusion for the grid — virtual-cathode region
where the electron fiow is toward the origin rather than away from it,
as was assumed in the derivation of the fundamental relations, it will
be convenient to change the symbol for transit angle from ^ to — f.
This will automatically take care of all algebraic signs, currents and
velocities now being considered positive when directed towards the
origin.

Since we are computing the impedance between an origin at the
virtual cathode and the grid plane we may apply (24) to find the po-
tential difference, getting

VACUUM TUBE ELECTRONICS 81

where Vi is the potential at the virtual cathode.
This relation is of the form

V„- V,^-{M+iN) (^'^,) [(l-cosf)-i-(f-sinf)] + /,Z„ (35)

where Jp is the plate current, and Z^ is the effective impedance:

In terms of the cold capacity Ci between plate and grid plane this
becomes

Z = -^^

PC. f ^

\^' -|(1 -cosr) +2sinf

which is plotted in Fig. 10.

The form of (35) shows that the equivalent network between the
plane of the grid and the plate may be represented by an equivalent
generator acting in series with the impedance, Z^. This is evidenced
by the fact that the velocity M + iN with which the electrons pass
the grid, may be expressed in terms of the grid potential Vg by means
of conditions between the grid and cathode. When complete space
charge exists near the cathode, these conditions are expressed by (25)
and (26). On the other hand, tubes with positive grid are sometimes
operated with inappreciable space charge between grid and cathode.
In this event, a similar analysis leads to values for the alternating-
current velocity and potential at the grid as follows:

U, = .1/ + iN = - |, I ( '' r '' ) + M ^—i^^ ) , (37)
V, = i^A =^, (38)

p pc

where 77 is the transit angle in the absence of space charge, and C is
the electrostatic capacity between unit area of cathode and of grid
plane. The right-hand side of (38) does not contain a minus sign be-
cause of the assumed current direction which is away from the cathode,
as is also the convention employed in (25) and (26) where the electron
charge e is a positive number.

The relations given by (35) allow the potential difference between
grid and plate to be determined in terms of the total current flowing
to the plate, and the total current flowing from the cathode, which ap-

[(^^^)+'(

— cos 7) \

V }\

Alvx . iA

i A = -^,

P pc

BELL SYSTEM TECHNICAL JOURNAL

pears in the velocity factor M + iN. In the usual case some of the
alternating current flows to the grid wires and is returned through an
external circuit connected to the grid. If the impedance between grid
and plate is desired it is necessary to find the relation which this grid
current bears to the total cathode and plate currents and to the alter-
nating-current potentials. The calculations involved are extremely
complicated because the assumption of current flow in straight lines
between parallel planes is far from representing the actual conditions

^P = -pC,^

Q9
Q8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

0 1 2 3 4 5 6 7 8 9 10 11 12

Fig. 10 — Plate impedance of positive grid triodes with slightly positive plate.

in the immediate neighborhood of the grid wires. Rather than at-
tempting an analysis of these conditions at the present time, we shall
content ourselves with results already obtained, since they are appli-
cable to the special case, which can be realized approximately in experi-
ment, where the grid is connected to a radio-frequency choke coil of
sufficiently good characteristics to prevent it from carrying away any
alternating current. For this special case the current J\ is the same
both in the cathode region and in the plate region, and all encumbering
assumptions involving different paths for the conduction and displace-

VACUUM TUBE ELECTRONICS 83

merit components of the current in the neighborhood of the grid wires
have been done away with.

The appHcation of the equations to this special case is dealt with
in the section of this paper devoted to positive grid oscillators. Be-
fore these oscillators can be treated comprehensively, a further exten-
sion of fundamental theory is necessary. This extension comes about
because positive grid oscillators are often operated with a slightly
negative potential applied to the plate.

VI. Triodes with Positive Grid and Negative Plate ^

When consideration is directed to tubes operating with positive
grid but negative plate, the fundamental underlying theory must again
be investigated. The reason for this lies in the fact that all electrons
which penetrate through the meshes of the grid are turned back be-
fore they reach the plate, so that in the grid-plate space there are two
streams of electrons moving in opposite directions. The effect of this
double value for the velocity may readily be calculated in so far as
direct-current components, only, are concerned. We have merely to
note that the charge density is double the value which it would have
in the presence of those electrons which are moving in one direction,
only, so that the correct relations are obtained from the equations al-
ready derived by taking twice the value of direct current in one
direction.

When alternating-current components are considered, however,
matters are more complicated, but not difficult. To see what the
actual relations are, let there be two possible values at any point for
the instantaneous velocity, and call these two values Ua and Ub, respec-
tively. Then the relation between force and acceleration becomes

eE^dUa^dlh . .

m dt dt ' ^ ^

Hence, at a given value of x we have by integration

Ua = Uh -\- constant.

But, when both values of velocity are separated into their components
according to (5) we have from (39)

Uao -\r Uai -\- • • ■ = Ubo -\- Ub\ + • • ■ + constant.

^ Since the publication of this paper in the Proceedings oi the Institute of Radio
Engineers, several questions have l)een raised regarding the treatment presented in
sections VI and VII. These are being investigated and will form the basis of
another paper.

84 BELL SYSTEM TECHNICAL JOURNAL

By equating corresponding terms, we find

UaQ = Ubo + constant,

Ual = Ubl,

Ua2 = Ub2, etc.

(40)

The first of these equations is trivial when the boundary conditions
are inserted, for then it appears that Uao = — Ubo and the equation
merely states that at a given value of x the direct-current velocity
component is not a function of time.

The second equation is much more enlightening and tells us that
although two values of the direct-current velocity may be present,
nevertheless there is only a single value for the alternating-current
component. The same conclusion holds for the higher order velocity
components. This conclusion supplies the key for the solution of the
general equations when applied to the stream of electrons moving in
both directions between the grid and plate of the tube.

In general, the total current may be written

1 f)F

J = PMa + PbUb+^ ^' (41)

47r at

If 2 is the total area of each of the electrode planes and

^ = a + b,

where a and b are constants to be defined later, (41) may be written
as follows:

In this expression, the two streams of current are clearly separated if
a and b are taken so that '^

Pa- = P and Pbj = P, (43)

a 0

where P is the total charge density, equal to the sum of P„ and Pb-

The total current may now be expressed in terms of velocities,
only, giving similarly to the transition from (1) to (3),

47r - 72 = a ( t/„|- + -^- )' U„ + b I Ub f + |: \Ub. (44)

m \ dx dt / \ dx at /

" A more rigorous analysis, involving nu-an values of the motions of individual
electrons, leads to the same result.

VACUUM TUBE ELECTRONICS

When Ua and Ub are each separated into their components accord-
ing to (5), so that (44) may be resolved into a system of equations,
we have for the first two equations, analogous to (6) and (7),

47r-/oS = (a - h)
m

ox \ ax

(45)

and

ox \ ox ox / di~ dx \ dx

+ (a-b)

dx \ dt / dt\ dx dx

, (46)

where the components of Ut have been expressed in terms of those of
Ua by means of (40) and the relation that Ubo = — Uao-
The solution of (45) is, as before,

where

Uo = a.r-'^

= IStT — /oa —

\ m a

(47)

Before attempting to solve (46) we make a change of variable as in
(10), writing

^^M^

and

U,=

This gives from (46)

P' d'-o: 1 3-0)

^ ^ ^ ^ d^dt j

(48)

In finding a solution for this, we shall restrict ourselves to the case
where all of the electrons turn back at the virtual cathode, so that
a = b and therefore the last term of (48) vanishes. The solution of
the remaining equation is then,

f/i

sin pt + \ F,{i^ + pt) + 7 F^iii - pi)

(49)

which is analogous to (12).

Again, assuming the two arbitrary functions to have the form,

Fiii^ + pt) = a sin (z^ -}- pt) + b cos (/t + pt),
Pikik - pi) = c sin {i^ - pt) -\- d cos (/^ - pt),

(50)

86 BELL SYSTEM TECHNICAL JOURNAL

and inserting the boundary conditions, (14) and (15), we have, in com-
plex form,

£/, = (.« + iN) -I a^ - 1-, ( 1 - f; aJi|). (51)

^ smh ^1 ^- \ ^ smh |i /

which is a simpler equation than its analogue (18). The potential is
obtained as in (24) giving.

]', = -

- (i1/ + iN) -——r [^ sinh t + 7(^ cosh ^ - sinh ^)]
-e I smh ^1 -"

9p'e

__ ^1^ sinh ^
sinh ^1

+ M ^-J - • u . (^ cosh ^ - sinh i)
' ^ smh ^1

+ constant. (52)

The alternating-current potential difference between the grid and
the virtual cathode where all of the electrons are turned back may be
obtained immediately from (52). As before, the variable f will be sub-
stituted for ^ to show that the grid-plate region is considered, and
currents and velocities will be considered positive when directed
towards the origin at the virtual cathode. Thus, from (52)

yp~e

~j - r coth i' + i'

(53)

The velocity, (AI + iN) may be expressed in terms of the alternating-
current grid potential, Vg, so that the path between grid plane and
virtual cathode may be represented by an effective generator in series
with an impedance, as was done in (34), (35), and (36).

VII. Oscillation Properties of Positive Grid Triodes ^

The oscillation properties of the positive grid triode are next to
be investigated. In the usual experimental procedure, an external
high-frequency circuit is connected between the grid and the plate of
the tube. It is unfortunate that this particular arrangement greatly
complicates the theoretical relations. Accordingly, a slightly modified
experimental set-up will be considered. This modification consists in
connecting the external circuit between the cathode and plate of the
tube, rather than between grid and plate. Experimental tests have
shown that the modified circuit exhibits the same general phenomena

'" Loc. cit.

VACUUM TUBE ELECTRONICS 87

as the more usual one, the difference being mainly one of mechanical
convenience in securing low-loss leads between the tube and the
external circuit.

The modified circuit, then, will be employed for analysis, and the
assumption will be made that the necessary direct-current connections
are made through chokes which are sufficiently good so that it may be
considered that no external high-frequency impedance is connected
between either the grid and the plate, or between the cathode and the
grid.

It is easy to see that under these conditions there can be no high-
frequency current carried away by the grid. It follows that for plane-
parallel structures, the alternating-current density, /], will be the same
both in the cathode-grid region and in the grid-plate region. The ar-
rangement thus reduces the problem to the consideration of the single
current, /i, and the resulting potential difference between cathode and
plate.

There are several possible combinations of direct-current biasing
potentials. For the first of these, the plate will be supposed to be
biased at a potential sufficiently positive to collect all electrons which
are not captured by the grid on their first transit. Complete space
charge will be assumed both in the cathode region and in the plate
region.

Under these conditions, we have the grid-cathode potential differ-
ence given by (26) and the grid-plate potential difference given by (35),
where the velocity, M + iN, is given by (25). We can write,

V,- v. = (v,-v„) + {v,~ ig

(55)
= - [Eq. 35] + [Eq. 26].

It will be remembered that the current was assumed to be positive in
(26) when directed away from the origin, and positive in (35) when
directed toward the origin. Therefore, since the same current exists
in both regions, and they are joined together at the grid, the sign of
the current Jy remains the same in both (35) and (26), its direction
being from cathode to plate. The impedance looking into the cathode-
plate terminals may be obtained from (55) by dividing by the ampli-
tude A of Ji and reversing the sign of the result to correspond to a
current from plate to cathode. Letting

Zo = Ro + iXo (56)

represent the impedance looking into the cathode-plate terminals, we
can write the result as follows

Ro =

BELL SYSTEM TECHNICAL JOURNAL

2 .

( 1 + cos 7] sin r; | (1 — cos f)

12ro

sin r? cos v ) ({' — sin {')

+ (2 cos 7] sin rj — 2)

(57)

Xo = rj^ H 1 + cos r; sin r; j (f - sin f )

/ 2 . 2 \

+ sin 77 cos ?7 (1 — cos t)

\V V /

+
+

y^ -^(1 - cosf) +2sinf
77 -\- -pV^ ~ 2 sin 77+77 cos 77

(58)

where 77 is the transit angle from cathode to grid, f is the transit angle
from grid to virtual cathode at the plate, and ro is the zero-frequency
resistance which would be present in a diode having the grid-plate
dimensions, and the same operating direct-current voltages and current
densities which occur in the grid-plate region of the triode under con-
sideration.

Fig. 11 shows graphically the relation between Rq and Xo for a
wide frequency range, in terms of the reference resistance, ro. Curve A
is drawn for the hypothetical condition that 77 = ^, so that the tube is
exactly symmetrical about the grid. Actually such a condition could
not be attained, since the grid captures some of the electrons, leaving
fewer for producing space charge near the plate. The grid-plate
dimension would accordingly have to be increased in order to secure
the space charge, but this would cause the transit angle f to become
larger than 77. However, despite the fact that it does not correspond
to a physically realizable condition, curve A is nevertheless of use in
indicating the limit which is approached as the grid capture fraction is
made smaller and smaller.

Curves B and C correspond to values of grid-plate transit angle
equal respectively to two and three times the cathode-grid transit
angle. Both these curves represent conditions which may readily be
obtained experimentally, and indeed, curves lying much closer to A
than does the curve B may be secured. For example, the general
relation for the ratio of the transit angles in terms of the direct currents
Ja and Jb in the cathode and in the plate region, respectively, when

VACUUM TUBE ELECTRONICS

complete space charge exists in both regions, is,

r ^

Suppose that the grid captured half of the electrons. Then the ratio
of transit angles would be 1.41. This would result in a curve lying
between A and B in Fig. 11.

The numbers, w/l, ir, and so forth, which are attached to the curves
in Fig. 11 show the values of the grid-plate transit angle, f, which
correspond to the points indicated.

RESISTANCE
-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

^ -0.6

u -0.8

4V
\

,4-TT

//c

4-n-*

'^a-TT

^1-

2-rT^

"^V

Fig. 11 — Ro — Xq diagram for positive grid, slightly positive plate triode with

cathode space charge.

Curve A, r) = ^
Curve B,-n = l/2f
Curve C, 77 = 1/3^

We now come to the problem of obtaining information about the
oscillation properties of a tube from a set of curves such as those shown
in Fig. 11. In a very loose way, and without proof we may state the
results of an extension of Nyquist's ^ rule as follows:

If an i? — X diagram, which in general may include negative as
well as positive frequencies, encircles the origin in a clockwise direc-

'II. Nyquist, "Regeneration Theory," Bell S\s. Tech. Jour., Vol. 11, p. 126;
January (1932).

90 BELL SYSTEM TECHNICAL JOURNAL

tion, then the system represented by the diagram will oscillate when
the terminals between which the impedance was measured are con-
nected together.

Verification of this rule, together with further extension to more
general cases are expected to be discussed in a subsequent paper. For
the present, its validity will have to be accepted on faith, but with
the assurance that the applications employed in this discussion are
readily capable of demonstration.

Returning to consideration of the positive grid triode with complete
space charge on both sides of the grid, and a slightly positive plate,
whose R — X diagram is given in Fig. 11, we see at once that the dia-
gram does not encircle the origin as it stands. Of course only positive
values of frequency are included in the curves as they are shown. The
inclusion of negative frequencies (never mind their physical meaning)
would produce a curve which would be the image of the curve shown,
a reflecting mirror being regarded as a plane perpendicular to the
paper, and containing the i?-axis. The curve A, for instance, would
have its part corresponding to negative frequencies lying above the
i?-axis and forming an image of the part lying below. This is shown by
the dotted curve in Fig. 12.

It is obvious that the curve of Fig. 12 will encircle the origin or
not depending on what happens at infinite frequencies. However, the
slightest amount of resistance in the leads to the tube will be sufficient
to move the curve to the right and thus exclude the origin. This
means that no oscillations would be obtained if an alternating-current
short were placed between plate and cathode. The result, although in
accord with experiment, is not particularly useful. The important
thing is to find whether the curve can be modified by the addition of a
simple electrical circuit in such a way that the origin of the resulting
R — X diagram for the combination of tube and circuit is encircled in
a clockwise direction.

Suppose that a simple inductance is connected in series with the
plate lead, and the impedance diagram of the series combination of
tube and inductance is plotted. For this arrangement, the R — X
diagram of Fig. 12 would be modified as shown in Fig. 13. Here the
part of the curve corresponding to negative values of resistance has
been pushed upward until the origin is enclosed within a loop which
encircles it in a clockwise direction. It is therefore to be expected that
oscillations will result. As to their frequency, we can say that the
grid-plate transit angle must be at least as great as 27r for this particular
example. This follows by supposing a certain amount of resistance to
be added in series with the circuit. The effect of this resistance will

VACUUM TUBE ELECTRONICS

be to move the curves on Fig. 13 bodily to the right. The lowest fre-
quency which will just allow the origin to be included within the loop
when the series resistance is reduced to zero and the inductance is
adjusted, corresponds to a grid-plate transit angle of lir.

It must be remembered that the foregoing details apply only to
curve A of Fig. 11, and it has already been pointed out that curve A

,^—

— --

..^

'^V

tf^c

]

,f=0

(

}

(

/"^

Fig. i:

-Curve A of Fig. 1 1 together with the image corresponding
to negative frequencies.

represents a limit which can be approached in practice, only as the
grid capture fraction is made smaller and smaller. Curve B can well
be duplicated in experiment. For this case, the lowest frequency at
which oscillations may be expected is much higher than before, since
the transit angle must be equal to 47r before the resistance becomes
negative. Actually, conditions intermediate between the two curves
may be realized, so that from a practical standpoint the transit angle

BELL SYSTEM TECHNICAL JOURNAL

must be in the neighborhood of Stt before we may expect to secure
oscillations.

This would correspond to a frequency somewhat higher than is
often associated with this type of oscillation. It must be remembered
however, that the particular case considered was that of a tube with
its plate at a slightly positive potential, whereas the majority of the
experimental frequency observations were made with the plate either
slightly negative, or, if positive, adjusted so that a virtual cathode was
formed inside the tube, and many of the electrons w^ere turned back

■ —

""^-"^

y
^

v._^

\
\

^211

•

/■

—

Fig. 13 — Modification of Fig. 12 produced by added inductance.

before they reached the plate. The curves of Fig. 1 1 do not apply to
these cases.

Therefore, let us see what happens when the plate is operated at a
negative potential so that all of the elections are turned back before
they reach it. At the outset, it should be remarked that this condition
does not prohibit the presence of direct-current plate current after
the oscillations have built up to a finite amplitude. The analysis applies
to the requirements for the starting of ihe oscillations, only, so that

VACUUM TUBE ELECTRONICS 93

if the plate fluctuates in potential by a very small amount, as it does
for incipient oscillations, and hence does not become positive during
the alternating-current alternation, then no direct-current plate cur-
rent can occur when the plate is biased negatively. After oscillations
have built up to an appreciable amplitude, the presence of plate cur-
rent is not only possible, but is in fact to be expected.

We have at hand the mathematical tools with which to compute
our R — X diagram for the negative plate triode with complete space
charge near the cathode. Thus, instead of substituting (35) in (55)
we must substitute (53). Since complete space charge is still postu-
lated near the cathode, (26) and (25) are still applicable. The result is:

R, =

12ro

1 + cos 77 sin 77 If^

/ 2 2 \

- ( sin 77 cos 77 j (f 2 coth f - ^)

-f (2 cos 77+77 sin 77 — 2)

(59)

12ro

2 . 2

sin 77 cos 77 1 t-

77 77

-(- I 1 + cos 77 sin 77 j {^~ coth i' — f )

+ iW - r- coth f + r) + (77 + k^ - 2 sin 77 + 77 cos 77)

, (60)

and the corresponding diagram is shown in Fig. 14. Here the curve A
shows oscillation possibilities for transit angles as small as 3/27r, while
a much greater amount of resistance would have to be added to the
circuit in order to eliminate the negative resistance and so stop the
oscillations. In all, then, this method appears to be a better way of
operating the system than with the positive plate, and this conclusion
is substantiated by experimental observations.

As before, an increase in the grid capture fraction moves the oscilla-
tion region up to higher frequencies.

In both of the examples cited above, and represented by Figs. 11 and
14, respectively, complete space charge was assumed near the cathode.
The effect of decreasing the cathode heating current so that this charge
becomes negligible may be computed by employing (37) in place of
(25), and (38) in place of (26).

The resulting equations for a slightly positive plate are,

BELL SYSTEM TECHNICAL JOURNAL

Ro =

12ro

r]—sinr]\, /I— cost;., . ^.

(61)

Xn=-—T^l I (r-sinf)+ ) (1-cosf

r I

K'-T(l-cosf)+2sinf

+K<-M- (62)

RESISTANCE
-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

0.2

-0.2

411

-0.4

|1.-

V--3TI^2

-0.6

? y

411

-a8

-1.0

"— V.

■zv

Z -1-2

^217

O —1.4
<

a. -1.6
-1.8

-2.0
-2.2
-2.4

■n

■3 A

Fig. 14 — Ro — A'n diagram for positive grid, slightly negative plate triode with

cathode space charge.

Curve A, T] = i;
Curve B,v = 1/2?
Curve r, 17 = 1 /3f

The corresponding R — X diagram is given in Fig. 15.

Again, the equations for a negative plate and no cathode space
charge are,

R, =

12ro

77 - sin r; \

1 — cos rj

(r- coth r - r)

, (63)

12r„

1 - cos r? \ .,, ^ ^ ,_-_sm, ^ ^^, ^^^^ ^ _ ^^

VACUUM TUBE ELECTRONICS

+ iU' -f-cothr + f) + hf^],
and the R — X diagram is shown in Fig. 16.

(64)

-0.2 0 0.2 0.4

4'rr

41T

^\r

2TT

2TT-

-IT

TT-

RESISTANCE
-t

0.2

-0.2

-0.4

-0.6

-0.8

-1.0

-1.2

-1.4

<J -1.6

I -1.8
(J

^-2.0
-2.2
-2.4
-2.6
-2.8
-3.0
-3.2
-3.4
-3.6

Fig. 15 — i?o — A'o diagram
for positive grid, slightly
positive plate triode, with-
out cathode space charge.

Curve A, r} = ^
Curve B,-n = l/2f
Curve C, Tj = l/3f

4TT-

^4 IT

4-rT-

-jS

-211

2Tr

\::

-TT

-TT-^

IT
- 2

TT-

IT
' 2

Fig. 16 — Ro — A'o diagram
for positive grid, slightly
negative plate triode, with-
out cathode space charge.

Curve A, 7) = ^
Curve B,-n = l/2f
Curve C, r; = l/3f

Inspection of Figs. 15 and 16 shows that the negative plate condi-
tion is greatly to be preferred when there is no cathode space charge.
In fact, when account is taken of the difference in the scales for which
Fig. 16 and the other three figures, 11, 14, and 15, are plotted, it is
evident that the negative plate without space charge offers the greatest
latitude in the adjustment of circuit condition. As in all of the cases,
except Fig. 15, a small grid capture fraction is to be desired. If

96 BELL SYSTEM TECHNICAL JOURNAL

curve A in Fig. 16 could be attained practically, it would be possible to
secure oscillations even at low frequencies by connecting an inductance
between the plate and cathode terminals. Curve B shows a low-fre-
quency limit of a little less than f tt for the grid-plate transit angle.

One of the more important observations to be drawn from the
curves of Figs. 11, 14, 15, and 16 is that if the inductance between plate
and cathode is obtained by means of a tuned antiresonant circuit, then
the circuit must be tuned to a frequency somewhat higher than the
oscillation frequency. This is in order that it may effectively present
an inductive impedance to the oscillating tube, so that the extended
curves in the figures may encircle the origin in a clockwise direction.

Another conclusion is that there are so many different permutations
and combinations of the operating conditions that it is small wonder
that there have been a great many different "theories" and empirical
frequency formulas advocated. For instance, operation under condi-
tions giving din R — X diagram which shows negative resistance over
a small frequency range, only, such as A in Fig. 11, or B in Fig. 15,
would give oscillations whose frequency would be much more nearly
independent of the tuning of the external circuit than would conditions
which resulted in a negative resistance over a wide frequency range,
as at A in Fig. 16. In this latter case the external circuit exerts a large
influence upon the frequency.

The data from which Figs. 11 to 16 were plotted are given in the
appended tables. The final step in the calculation of these data was a
multiplication by 12 which was performed on a slide rule. For all
previous steps seven-place tables were employed because of the fre-
quent occurrence of differences of numbers of comparable magnitude.

The effect on the frequency of a change in the operating voltages
can be deduced inferentially from the curves. Thus, in general, the
formulas for the transit angle have the form,

\\Vo

where x is the grid-origin distance,
X is the wave-length,
V[) is the grid potential,
i^ is a constant which depends on the mode of operation.

When the plate potential is changed by a relatively large amount
the operation undergoes a transition from a limiting mode illustrated
by one of the figures to another limiting mode shown on some other
one of the figures.

On the other hand, a change in grid potential will act to change the

VACUUM TUBE ELECTRONICS

transit angles on the two sides of the grid in the same proportion. A
modification of this generaHty occurs because the value of x in (65)
will shift as the effective position of the virtual cathode moves about.
Also, the complete space-charge condition near the plate becomes
modified, and so the general relations become extremely variable. The
partial space charge that exists with very negative values of plate
potential, or with very high values of grid potential does not lend itself
readily to mathematical treatment, so that intermediate conditions
between complete and negligible space charge can be treated only by
inference as to what happens between the two limiting conditions.

With inappreciable space charge on both the plate and the cathode
sides of the grid, there can be no oscillations at all, since all impedances
then approach pure capacities, with no negative resistance components.

A word concerning the so-called "dwarf" waves is in order before
this general theoretical discussion is completed. In the curves. Fig. 14
distinctly shows this possibility in curve A , since the resistance reaches
a large negative value at 2ir and again at 47r. Likewise Fig. 1 1 shows
the same possibility. On account of the resulting confusion in the
figures, the higher frequency portions have not been drawn in the
figures, but from (57), (59), (61), and (63) we can see what happens.
Thus, for very high frequencies, t] is large compared with unity, so
that the formulas may be written.

Ro= -

— -(t/ + 0 smr],

(57-a)

Rq = ^ (1 + cos 7/ + sin 7}),

R,= -

i?n -

12ro
12ro
12ro

1 + xl sin ( r; + ^ j

1 — - cos ^
V

-cos 17

(59-a)

(61-a)

(63-a)

It is noteworthy that all of these exhibit the possibility of "dwarf"
waves separated by discrete frequency intervals except (63-a). On the
other hand, (63-a) gives possible conditions for operation at all high
frequencies provided that the proper external circuit may be secured.

VIII. Postscript

The extension of the electronics of vacuum tubes which was de-
scribed in the preceding pages must be regarded in the light of a tenta-

98 BELL SYSTEM TECHNICAL JOURNAL

tive starting point rather than as a completed structure. Of the funda-
mental correctness of the method of attack there can be Httle doubt.
The various simpHfying assumptions, however, require careful scru-
tiny and doubtless some of them will be revised as time goes on and
additional experience is acquired. Experimental guidance will be in-
valuable, and indeed certain data already have been obtained which
are helpful in analysis of the assumptions. Although these data are in
general qualitative agreement with the theory as outlined, the ex-
perimental technique must be refined before quantitative comparison
can be made. It is hoped that the results can be made available at an
early date.

Among the various assumptions which were made in the develop-
ment of the theory, there are three which lead particularly to far-
reaching consequences. These three may be enumerated as follows:

1. Plane-parallel tube structures

2. Current flow in straight lines

3. Small alternating-current amplitudes.

There are grounds for the belief that the assumption of plane-
parallel tube structures does not exclude the application of the alter-
nating-current results to cylindrical structures as completely as might
be supposed. In the first place, the approximation of cylindrical
arrangements to the plane-parallel structure becomes better as the
cathode diameter is made large. Many tubes contain special cathode
structures where this is the case. Furthermore, Benham ^ has obtained
an approximate solution for the alternating-current velocity in cylin-
drical diodes where the cathode diameter is vanishingly small, and the
transit angle is less than 5 radians. The resulting curves of alternating-
current velocity versus transit angle have the same shape as the curves
for the planar structures, and when the cylindrical transit angle is
arbitrarily increased by about 20 per cent, the quantitative agreement
is fair for transit angles less than 4 radians. It follows that until
accurate solutions for cylindrical triodes can be obtained, the planar
solutions may be expected to give correct qualitative results, and fair
quantitative results when appropriate modifications of the transit
angle are made. In fact, good agreement is obtained if calculations of
the cylindrical transit angle are made as though the structure were
planar.

The assumption of current flow in straight lines is open to some
question when a grid mesh is interposed in the current path. For the
positive grid triode, the objection to the assumption has been over-
come by postulating a special case where an ideal choke coil prevents

VACUUM TUBE ELECTRONICS 99

the grid from carrying away any of the alternating current. Benham ^
has suggested an alternative which seems to work fairly well when the
grid-cathode path of a negative grid tube is considered, but which offers
grave difficulties when the grid-plate path is included. A still different
alternative was employed in the present paper in connection with
negative grid triodes, and successfully indicates phase angles for the
mutual conductance of the tube which are qualitatively logical. The
grid-plate path is still without adequate treatment, however.

As to the third general assumption : that of relatively small alternat-
ing-current amplitudes, there can be no objection from a strictly math-
ematical point of view, and for a very large proportion of the physical
applications the assumption is thoroughly justified. Indeed, it is the
only one which is successful in giving starting conditions for oscillators.
However, when questions as to the power efficiency of oscillators or
amplifiers arise, then the "small signal" theory is inadequate, and
should be supplanted by an approximate theory. The form which this
approximate theory should take is indicated by the standard methods
of dealing with the efficiencies of low-frequency power amplifiers and
oscillators where the wave shape of the plate current is assumed to
be given. The application of the same kind of approximation to ultra-
high-frequency circuits may eventually prove to be a simpler matter
than the "small signal" theory set forth in these pages.

Besides the three main assumptions discussed above, there was a
fourth assumption which, although of lesser importance, deserves some
comment. This fourth assumption involves the neglect of initial veloc-
ities at a hot cathode. If all electrons were emitted with the same
velocity, the theory is adequate, and may be applied as indicated by
Langmuir and Compton.' When the distribution of velocities accord-
ing to Maxwellian, or Fermi-Dirac, laws is considered, some modifica-
tions may be necessary. In general, a kind of blurring of the clear-cut
results of the univelocity theory may be expected, which will be
expected to result in an increase in the resistive components of the
various impedances at the expense of the reactive components. Again,
lack of symmetry in the geometry of the tube structure may be ex-
pected to do the same thing, since the transit angles are then different
in the different directions.

Finally, however, and with all its encumbering assumptions, it is
hoped that the excursion back to fundamentals which was made in
this paper, has resulted in a method of visualizing the motions of the
condensations and rarefactions of the electron densities inside of
vacuum tubes operating at high frequencies and has shown their rela-
tion to the conduction and displacement components of the total
current.

100

BELL SYSTEM TECHNICAL JOURNAL
Data for Fig. 11

V =r

V =

1.0

1.87

-0.577

0.302

-0.119

0.0633

1.4

1.75

-0.777

0.292

-0.164

0.0604

-0.160

1.57

1.69

-0.855

0.288

-0.182

0.112

-0.172

1.8

1.60

-0.949

0.280

-0.204

0.108

-0.193

0.0568

-0.200

2.356

1.36

-1.13

0.260

-0.241

0.0987

-0.240

2.8

1.15

-1.23

0.241

-0.289

0.0458

-0.278

3.14

0.986

-1.27

0.226

-0.311

0.0838

-0.289

0.0418

-0.297

3.6

0.769

-1.29

0.204

-0.335

0.0748

-0.308

0.0364

-0.316

4.0

0.593

-1.27

0.186

-0.350

0.0673

-0.320

0.0320

-0.327

4.71

0.326

-1.18

0.153

-0.366

0.0457

-0.329

5.2

0.186

-1.08

0.132

-0.368

0.0216

-0.330

5.6

0.0972

-0.987

0.116

-0.368

0.0193

-0.324

6.28

-0.830

0.0923

-0.358

0.0365

-0.309

0.0165

-0.306

6.8

-0.0348

-0.719

0.0767

-0.346

0.0153

-0.291

7.2

-0.0440

-0.644

0.0662

-0.337

0.0308

-0.284

0.0148

-0.278

7.85

-0.0369

-0.546

0.0515

-0.318

0.0284

-0.264

8.4

-0.0199

-0.487

0.0414

-0.302

0.0268

-0.248

0.0144

-0.238

9.0

+0.000414

-0.444

0.0320

-0.284

0.0254

-0.233

0.0144

-0.222

9.42

0.0125

-0.424

0.0262

-0.272

0.0244

-0.222

0.0143

-0.235

10.0

0.0218

-0.407

0.0194

-0.256

0.0138

-0.198

10.99

0.0213

-0.385

0.0101

-0.232

0.0192

-0.193

12.57

-0.342

-0.197

0.0128

-0.172

0.00962

-0.162

Data for Fig. 14

V =f

V =

§{•

V =

1.0

2.94

-0.932

0.581

-0.225

0.133

-0.221

1.4

2.84

-1.33

0.595

-0.304

0.136

-0.286

1.57

2.78

-1.49

0.601

-0.336

0.252

-0.302

1.8

2.67

-1.71

0.606

-0.377

0.255

-0.330

0.139

-0.336

2.356

2.30

-2.19

0.618

-0.465

0.264

-0.381

2.8

1.90

-2.48

0.619

-0.528

0.147

-0.401

3.14

1.56

-2.63

0.613

-0.571

0.272

-0.424

0.150

-0.410

3.6

1.06

-2.71

0.598

-0.625

0.276

-0.438

0.153

-0.416

4.0

0.645

-2.67

0.577

-0.665

0.277

-0.448

0.155

-0.417

4.71

0.00015

-2.38

0.525

-0.728

0.275

-0.460

0.157

-0.412

5.2

-0.319

-2.08

0.479

-0.762

0.158

-0.407

5.6

-0.494

-1.80

0.436

-0.784

0.158

-0.404

6.28

-0.608

-1.31

0.356

-0.807

0.253

-0.476

0.156

-0.396

6.8

-0.565

-0.990

0.292

-0.814

0.154

-0.390

7.2

-0.480

-0.796

0.241

-0.803

0.232

-0.483

0.152

-0.386

7.85

-0.290

-0.594

0.160

-0.797

0.213

-0.485

8.4

-0.111

-0.528

0.0957

-0.773

0.196

-0.486

0.144

-0.376

9.0

+0.00226

-0.536

0.0308

-0.735

0.175

-0.485

0.138

-0.372

9.42

0.0574

-0.574

-0.0102

-0.705

0.160

-0.484

0.133

-0.369

10.0

0.077

-0.634

-0.0581

-0.666

0.127

-0.365

10.99

-0.690

-0.118

-0.564

0.101

-0.436

12.57

-0.152

-0.560

-0.152

-0.414

0.0445

-0.383

0.0938

-0.348

VACUUM TUBE ELECTRONICS
Data for Fig. 15

101

1 =r

V = k

V =

= K

V =

-\'o

1.0

0.239

-3.06

0.179

-1.58

1.4

0.228

-2.22

0.172

-1.19

1.57

0.223

-2.00

0.199

-1.39

1.8

0.215

-1.76

0.192

-1.24

0.162

-0.979

2.356

0.193

-1.39

0.173

-1.01

2.8

0.173

-1.21

0.131

-0.737

3.14

0.157

-1.10

0.130

-0.825

0.120

-0.693

3.6

0.135

-0.983

0.123

-0.756

0.104

-0.645

4.0

0.116

-0.903

0.0902

-0.610

4.71

0.0837

-0.788

0.0796

-0.633

5.2

0.0643

-0.724

0.0540

-0.524

5.6

0.0503

-0.677

6.28

0.0308

-0.605

0.0346

-0.506

0.0308

-0.455

6.8

0.0197

-0.558

0.0232

-0.425

7.2

0.0131

-0.525

0.0194

-0.444

0.0187

-0.402

7.85

0.00568

-0.477

0.0126

-0.406

8.4

0.00203

-0.442

0.00939

-0.378

0.109

-0.343

9.0

-0.0000284

-0.409

0.00710

-0.351

0.00908

-0.318

9.42

-0.000646

-0.388

0.00608

-0.333

0.00826

-0.290

10.0

-0.000817

-0.363

0.00744

-0.284

10.99

-0.000402

-0.327

0.00408

-0.282

12.57

-0.285

0.00216

-0.246

0.00385

-0.227

Data for Fig. 16

n =f

V =

= U

V =

= U

V =i

1.0

-0.176

-9.35

0.426

-4.84

0.342

-2.52

1.4

-0.306

-6.84

0.366

-3.64

0.316

-1.96

1.57

-0.363

-6.15

0.338

-3.33

0.345

-2.32

1.8

-0.436

-5.41

0.299

-3.00

0.321

-2.11

0.287

-1.67

2.356

-0.584

-4.15

0.206

-2.46

0.263

-1.77

2.8

-0.658

-3.44

0.137

-2.17

0.212

-1.29

3.14

-0.685

-3.00

0.0888

-1.99

0.188

-1.48

0.190

-1.21

3.6

-0.685

-2.52

0.0318

-1.80

0.149

-1.36

0.162

-1.12

4.0

-0.661

-2.17

-0.0104

-1.65

0.120

-1.27

0.140

-1.06

4.71

-0.565

-1.69

-0.0698

-1.43

0.0747

-1.13

0.108

-0.957

5.2

-0.482

-1.45

-0.0998

-1.30

0.0871

-0.898

5.6

-0.413

-1.30

-0.119

-1.21

0.0730

-0.853

6.28

-0.304

-1.11

-0.141

-1.07

0.0478

-0.907

0.0523

-0.787

6.8

-0.236

-1.02

-0.151

-0.975

0.0389

-0.742

7.2

-0.195

-0.963

-0.155

-0.910

-0.0220

-0.805

0.0296

-0.709

7.85

-0.148

-0.894

-0.156

-0.815

-0.0364

-0.743

8.4

-0.126

-0.848

-0.152

-0.747

-0.0458

-0.695

+0.0072

-0.625

9.0

-0.113

-0.803

-0.145

-0.680

-0.0538

-0.647

-0.00162

-0.589

9.42

-0.109

-0.771

-0.138

-0.638

-0.0582

-0.615

-0.00706

-0.565

10.0

-0.107

-0.727

-0.128

-0.588

-0.0135

-0.535

10.99

-0.101

-0.653

-0.107

-0.518

-0.0668

-0.517

12.57

-0.0760

-0.556

-0.0760

-0.439

-0.0667

-0.439

-0.0314

-0.426

Contemporary Advances in Physics, XXVII
The Nucleus, Second Part *

By KARL K. DARROW

In this Second Part the major subject is Transmutation: that is to say,
the alteration or disintegration of a nucleus, the unique and distinctive part
of any atom, by impacts of fast-moving corpuscles. For the last year and
a half the pace of progress in this field has been increasingly rapid, and in all
likelihood is destined to become yet swifter. This is partly because of the
discovery — a discovery due largely to theoretical foresight — that trans-
mutation of some elements is practicable with protons of a relatively
modest energy which can be produced in laboratories without any serious
difficulty. Partly it is due to the discovery of neutrons and of deutons,
particles which apparently possess remarkable ability in effecting certain
kinds of transmutation. Partly also it is due to advances and refinements
in the methods of working with alpha-particles, the first variety of corpuscle
with which disintegration of nuclei was ever achieved. People are already
beginning to speak of "nuclear chemistry" as a special branch of science,
and this is already almost justified by the number of cases known in which
two nuclei interact and produce two others which are recognizable.

Bringing the First Part up to Date

STRANGE as it seems to speak of "bringing up to date" something
that was pubUshed only six months ago, one is sometimes obHged
to do so by the rapid march of science; and three of the "elementary
particles" of which I spoke in the First Part were and still are so
young— or to speak more carefully, our acquaintance with them is
still so young — that their role and situation in the body of physical
knowledge is changing from month to month.

The Positive Electron
Of the positive electron the most striking new thing to be said is,
that there is now a new way of generating it: by impacts of alpha-
particles against metals. This so far has been applied only by its
discoverers, M. and Mme. Joliot; only with alpha-particles from
polonium, therefore of energy 5.3 millions of electron-volts; only to
five metals, of which beryllium and boron and aluminium yielded
positive electrons, while silver and lithium did not. It is as yet the
most efficacious way of producing positive electrons, Joliot having
evoked last summer as many as 30,000 of these corpuscles per second
from aluminium. This of course looks small when compared with the
torrents of negative electrons which incandescent metals will pour out,

* "The Nucleus, First Part" was published in the July 1933 issue of the Bell Sys.
Tech. Jour., Vol. XII.

102

CONTEMPORARY ADVANCES IN PHYSICS

103

but these are not a proper standard of comparison. Rather should
one say that in the autumn of 1932 positive electrons were being
observed at the rate of three or four a year, and already by the summer
of 1933 this rate had been enhanced to thirty thousand in the second!
The other voluntary way of generating positive electrons — -by
applying hard gamma-rays to heavy elements — ^has already been
studied enough to yield the data of the following table. Here, in the
first column, stand the names of various sources of gamma-rays (the
one denoted as "Po + Be" is beryllium exposed to impacts of alpha-
particles from polonium); in the second, the energy-values in MEV
(I use this symbol hereafter for "millions of electron-volts") of the
individual photons of these rays; in the third, the symbols of various
metals; in the fourth, the number of positive electrons per hundred
negatives, ejected from these metals by these gamma-rays; in the
fifth, the authorities:

Po-f Be

Joliots

Chadwick

Joliots

ThC"

2.6

Joliots

Chadwick

Ra(B -t- C)

1.0-2.2

Grinberg

0.85

Meitner-Philipp

The percentages in the fourth column give at the moment our best
available notion as to the relative plentifulness of positive electrons,
produced by the several kinds of rays falling upon the several metals.
One would prefer to have the total number of positives per unit
intensity of the infalling rays, but that is not available at present — -
I presume because of the difficulty of measuring these intensities.
One must remember that the data usually consist in observations of a
few hundred or a few dozen cloud-tracks, so that the accuracy of
these percentages cannot be great. ^

We note that with lead the proportion of positive electrons mounts
rapidly with increasing photon-energy, and that with 5 MEV-photons

^ This perhaps is sufficient to account for a discrepancy between the general trend
of the table and a value of 1/3 given by Meitner and Philipp for the ratio of positives
to negatives when brass is exposed to (Po + Be). Should the table be extended and
supported by a successful theory, it should then be possible to determine the frequency
of gamma-rays by the percentage of positives which they produce when falling on a
metal. In this connection it is interesting that Anderson's latest data indicate that
positive and negative electrons are about equally abundant among the ionizing
particles of the cosmic rays, a fact which suggests that if they are due to photons,
these must be of a distinctly higher energy than anv of those cited in the foregoing
table.

104 BELL SYSTEM TECHNICAL JOURNAL

the proportion goes up rapidly with the nuclear mass of the bombarded
atoms.^ Both of these rules are in harmony with the remarkable
theory to which I alluded in the First Part — ^the theory that each
positive electron (together with a negative companion) springs into
being from a transmutation of light into electricity! It is supposed
that a photon transmutes itself into a pair of electrons, one of each sign.

Fig. 1 — Tracks of an electron-pair (positive and negative) arising in argon exposed
to gamma-rays, and probably crekted near an argon nucleus by transmutation of a
photon. (M. et Mme. Joliot)

Conservation of the net charge of the universe is assured in this
hypothetical process. Conservation of mass and energy is attainable,
for the speeds of the electrons may be such that their energies together
are equal to the energy of the vanished photon. I take this occasion
to repeat Einstein's principle, which figures so importantly in these
articles. The energy £ of a material particle moving with speed v
(relatively to the observer) is given by the formula:

E = moil - |82)-i/2c2 = mc\

^ In this connection it should be noted that the source "Po + Be" emits neutrons
as well as photons, and while the first-named are certainly not chiefly responsible for
the positive electrons, they may produce some of these. Chadwick observed that
the rays from a " Po + B" source (boron in place of beryllium), which consist of
neutrons plus some photons of about the same energy as the photons of ThC",
evoked from lead a distinctly larger percentage of positive electrons than do the rays
of ThC".

CONTEMPORARY ADVANCES IN PHYSICS 105

in which c stands for the speed of light in vacuo; ^ for vjc; and Wo
for a constant. The ratio of E to c^ is the function of v and mo which
this equation defines, and is denoted by m and called the mass of the
particle: it is in this sense that mass and energy are equivalent.
We may (mentally) divide the mass of the moving particle into two
terms mo and {ni — mo), and the energy into two terms moC- and
(m — mo)c^. We may further call mo the rest-mass and moc"^ the
energy associated with the rest-mass; and we may call (m — n?o)c^
the kinetic energy and (m — mo) the mass associated with the kinetic
energy or the extra mass due to the motion of the particle. Such will
be the terminology used in these articles, although this definition of
kinetic energy is only approximately the same as the classical and
familiar one.^

Returning to the argument about the transmutation of light into
electrons, or more precisely, of a photon into an electron-pair : conserva-
tion of mass and energy is attainable, for the two electrons may have
such speeds — call them ^\C and ^iC — that the sum of mo(l — ^i')~'^''^c'^
and mo(l — ^'^)~^^H^ is equal to the energy hv of the photon. But
the demand for conservation of momentum makes apparently serious
trouble. If we assume that a photon voyaging through the depths of
space suddenly converts itself spontaneously into a pair of electrons,
and if then we attempt to impose both conservation of momentum
and conservation of energy, the equations lead us straightway into an
inescapable muddle, in which the original assumptions contradict each
other. We are driven therefore to infer that the imagined process is
impossible. But this seeming catastrophe of the theory turns out to
be a blessing. What is observed is not after all the transmutation of
a photon in the depths of empty space, but a process which occurs in
the depths of plates of lead and other heavy elements. If we suppose
that such a transmutation occurs near to a massive nucleus, then this
may receive some of the energy and some of the momentum of the
photon; and the equations show that the momentum which it takes
may be quite sufficient to permit the process to occur, while the
energy which it takes is so small that for practical purposes we may
still pretend that the whole of the energy of the photon is divided
between the electrons (though we certainly should not forget about
the small fraction which goes to the nucleus). All the principles are
thus fulfillable: conservation of charge requires that there should be

^ The classical definition of kinetic energy is {'\./2)inv^; the present or relativistic
definition, viz. {ni — mo)c^, is an infinite series of which the first term is identical with
the classical definition. The difference between the two definitions increases as the
speed of the particle increases, but so far as I know there has not yet been an actual
case in which it is of practical importance.

106 BELL SYSTEM TECHNICAL JOURNAL

two electrons of opposite signs; conservation of energy, that they
should have appropriate speeds; conservation of momentum, that the
process should occur only near a massive nucleus.

This is the most alluring of all theories, for it is the doctrine that
the substance of matter and the substance of light are ultimately the
same, being interconvertible. It therefore demands, and is surely
destined to receive, the sharpest and fullest of testing; the more so
because there is a rival in the field, the theory that the positive electron
exists beforehand and from all time in the nucleus of the atom, and is
ejected from it by the photon. The newest way of producing positive
electrons by alpha-particle impact seems to speak in favor of the latter.
One could indeed suppose that the kinetic energy of the alpha-particle
is transformed into an electron-pair, directly or through the inter-
mediacy of a transiently-existing photon ; but this would be an artificial
idea unless it were to be supported by a basic theory or by observing
that the positive electrons are often paired with negatives (which the
Joliots do not say). In favor of the former theory speak the facts
that in several scores of cases paired electrons have been observed —
i.e., two electrons of opposite signs were seen to spring from the same
point (so far as the eye could tell) — when metal plates were bombarded
with gamma-rays; and the further fact that the energies of these
electron-pairs and of individual positives did not surpass those of the
infalling photons, though they approached it often. ^ There are always
apparently unpaired positives and many more unpaired negatives; but
one may always say that with some of the pairs it happened that one
member remained in the metal and the other got away, while many of
the negatives are surely electrons which have been expelled from their
places by photons acting in the well-known ways. Further, there are
more or less forcible indications that some part of the absorption of
gamma-rays in heavy metals may be ascribed to the formation of
electron-pairs, and some part of the radiation scattered from the
metals when gamma-rays fall on them may be attributed to the
reunion of two electrons of opposite sign which re-transmute them-
selves into light; but some of the data are not checked, and the time
seems not ripe for reviewing them. In the hands of Oppenheimer
and Plesset the transmutation theory has supplied other quantitative

^See First Part of this article, pp. 304-305, B.S.T.J., July 1933. The kinetic
energies of electron-pairs and a fortiori of positive electrons should not come within
one million electron-volts of the energy-value of the photons, for the rest-mass of
two electrons amounts approximately to a million of these units. This rule has
lately been strengthened by evidence from Anderson and his colleagues, who in a
couple of hundred of additional cases find no violation of it; the distribution-in-energy
curves for pairs and for (apparently) isolated positives extend up to the predicted
upper limit, and there they fall to the horizontal axis. More evidence of this kind
has been accumulated by Blackett (loc. cit. footnote 5).

CONTEMPORARY ADVANCES IN PHYSICS 107

predictions meet for testing, and it is likely that in six months more a
great deal will be learned.^

The Denton

The newly-discovered isotope of hydrogen of mass-number 2 — H',
"heavy hydrogen," or, to adopt Urey's name for it, "deuterium"-^—
has suddenly become the most popular and the most eagerly sought-
after of all chemical substances. This is because of the notable
chemical and physical differences between it and its compounds on
the one hand, H^ and the corresponding compounds of H^ on the other.
So great are these differences that by the usage of twenty years ago
H^ would probably have been called a new element, and indeed it
deserves all the prestige that would accrue to it from being so denoted ;
but to violate the present and most wisely-based of usages, whereby
an element is characterized by atomic number rather than by the
ensemble of its propeities, would be mistaken.''

Deuterium is so rare by comparison with H^ (Urey's "protium")
that it would still be very unfamiliar, but for the unexpected and
remarkable efficacy of the electrolytic method of separating water
molecules comprising H^ atoms from water molecules comprising none
but H^ atoms. It turns out that if an aqueous solution is electrolyzed
until only a very tiny fraction of the original liquid remains, the
proportion of the former kind of molecule in that tiny residue is
anonialously large. Washburn seems to have been the first to suspect
that this might happen; he procured samples of the residues from
electrolytic cells which had been operated continuously in commercial
plants for two and three years, and sent them to Urey, who performed
a spectrum-analysis and observed "a very definite increase in the
abundance of H^ relative to H^." Shortly afterwards the method was
put into operation on a grand scale by G. N. Lewis and his collabora-
tors, with spectacular results. In one experiment, for instance, they
started out with twenty liters of water, electrolyzed it until there
remained but half a cc. of liquid, and found that in this residue deu-
terium atoms made up two-thirds of all the hydrogen atoms which
were left. For months thereafter, nearly every paper on deuterium
and on the deuton which was published began with an acknowledgment
to Lewis for a small amount of water rich in heavy hydrogen which the
fortunate author had received from him.

^ For a fuller account of the situation as it now stands, see an article of mine in the
Scientific Monthly, January 1934; also one by P. M. S. Blackett, Nature 132, pp. 917-
919 (Dec. 16, 1933), which incidentally contains some further data.

" I should think that the case of deuterium by itself would make it necessary
henceforth to define the concept "element" altogether from the concept "atomic
number," forsaking all the earlier definitions.

108 BELL SYSTEM TECHNICAL JOURNAL

Interesting as are the chemical and physical properties of deuterium
and its compounds, we are here concerned only with the nucleus of the
H^ atom, the deuton (all the other suggested names seem to be fading
out). The accepted value for its mass is that given by Bainbridge,
2.0131 on the standard scale in which the mass of the O^^ atom is
16 exactly. Of its spin I shall speak in a later article. Its powers of
transmutation are remarkable, and quite unlike those of H^; if first a
beam of H^ nuclei (protons) and then an equal beam of deutons be
directed against targets of various elements, the number of fragments
observed per unit time is greater for some elements and less for others,
and their ranges in general are different. In some cases it seems possible
that the deutons themselves are being split into protons and neutrons,
a result of great importance if it can be established beyond question.
We shall consider the data at length.

The neutron

Most of what has newly been learned about the neutron will find
appropriate places elsewhere in this article. There should be a
separate section about the deflections suffered by neutrons when they
impinge on or pass close to nuclei without transmuting them — the
topic known as "scattering," "interception," or (badly) "absorption"
of neutrons. This topic however is scarcely ripe for description in
such an article as this, the experiments being difficult and the in-
ferences from the data being highly controversial. I therefore post-
pone it to some future occasion, remarking only that it seems established
that a neutron may pass within a very short distance indeed from a
nucleus — only a very few times 10~^^ cm from the centre thereof —
without interacting with that nucleus in any perceptible way.

Masses of the Lighter Atoms

There are now thirteen of the lighter atoms of which the masses —
in terms of the mass of the O'^ atom taken as 16 exactly — have been
determined to four and even to five significant figures. Most of these
values were mentioned in the First Part, but it will be convenient to
have them all tabulated here. They are the masses of complete
atoms, nuclei accompanied by their full quotas of orbital electrons.
The uncertainties quoted are the "probable errors"; where Aston
originally gave the maximum" possible uncertainty, this has been
divided by 3 (see First Part, footnote 10). Values marked with an
asterisk are from Bainbridge, the others from Aston; the value for H^
has been obtained by both.

CONTEMPORARY ADVANCES IN PHYSICS 109

W 1.007775 ± .000035 C^^ 12.0036 ± .0004

*H2 2.01363 ± .00008 N" 14.008 ± .001

He4 4.00216 ± .00013 O^^ 16.0000 (standard)

*Li« 6.0145 ± .0003 F^^ 19.000 ± .002

*Lr 7.0146 db .0006 * Ne^o 19.9967 ± .0009

*Be' 9.0155 ± .0006 * Ne^^ 21.99473 ± .00088

Bi" 10.0135 ± .0005 *CF5 34.9796 ± .0012

B" 11.0110 ± .0005 *CF 36.9777 ±'.0019

The table of the chemical atomic weights reproduced in the First
Part has suffered two alterations: a very slight change in the given
value for K, from 39.10 to 39.096; and an important change in the
chemical atomic weight of carbon, which rises from 12.00 to 12.011,
and now permits of an abundance of C^^ easier to reconcile with the
observed intensities of the spectrum-lines of this substance than was
the abundance, or rather the scarcity, implied by the former value.^

The list of isotopes detected by Aston 's mass-spectrograph has been
enlarged by the following examples,^ which the reader may enter upon
Fig. 6 of Part I: neodymium, Z = 60, A-Z = 83; samarium, Z = 62,
A-Z = 85, 86, 87, 90, 92; europium, Z = 63, A-Z = 88, 90; gadolin-
ium, Z = 64, A-Z = 91, 92, 93, 94, 96; terbium, Z = 65, A-Z = 94.

New Developments in Transmutation: The Apparatus

In the two years and a quarter which are all that have elapsed
since I published in this Journal an article on transmutation,^ the
situation in this field has vastly changed, and the prospects for the
future have been amplified immensely. So lately as the early spring
of 1932, disintegration of a nucleus had not yet been demonstrably
achieved except by alpha-particles possessing energy not smaller than
three millions of electron-volts. Schemes for producing five- and ten-
million-volt ions were already under way, being ardently pushed
onward because it was supposed that transmutation would never be
effected by any agency much feebler. But in the course of 1930,
Cockcroft and Walton of the Cavendish Laboratory had been em-
boldened by a theory (I will describe it later) to imagine that protons
of only a few hundred thousand electron-volts might be able to
transmute, and to risk their time and labors in the task of developing
powerful streams of such particles. After two years of work they

' See an item in Nature, 132, 790-791 (Nov. 18, 1933). In the table of masses on
p. 303 of the First Part, change 1.0078 to 1.0072 and 4.002 to 4.001 (the former
values refer to complete atoms, not bare nuclei).

* F. W. Aston, Nature 132, 930-931 (Dec. 16, 1933).

^ "Contemporary Advances in Physics XXII," Bell System Technical Jotirnal, 10,
628-665 (October 1931). I refer to this article hereinafter as Transmutation.

no BELL SYSTEM TECHNICAL JOURNAL

were justified in the event; for they detected fragments proceeding
from targets of lithium bombarded by their protons, with energy-
values anywhere from half-a-million down to only seventy thousand
electron-volts.

It would be hard to overstate the joyful surprise of this announce-
ment. Transmutation, of some elements at least, was easier by far
than had been thought! It would not after all be necessary to fare
forth into the unknown, and face at once the problems of applying
voltages without precedent; successes which had seemed doubtful at
best and assuredly distant were after all to be had by a relatively
slight extension of a known technique. All over Europe and America
people began making plans for applying these voltages, so much less
formidable than those which had previously been thought indispen-
sable. Nevertheless the first who confirmed and extended the work
of Cockcroft and Walton were those who had aimed from the start
at the higher and harder goal : Lawrence and his colleagues at Berkeley.
Their work had not been wasted, for they instantly found themselves
able to measure the disintegration of lithium by protons all the way
up to 710,000 electron-volts; and within four months they had carried
the upper limit onward to 1,125,000, and as I write these lines they
have just announced that the limit has soared to three millions!
From Pasadena also comes word of transmutation achieved by protons,
and deutons, and helium nuclei, endowed with energy by voltages
ranging downward from nine hundred thousands.

These are not the only novel results of the last two years and a
quarter. The neutron has disclosed itself not only as a product, but
as an agent of transmutation, able to alter nuclei which have thus far
resisted both the alpha-particles and the protons which have been
showered upon them in laboratories. The disintegrations effected by
alpha-particles have been studied with ever-increasing minuteness
and detail, and are beginning to show that nuclei are structures
capable of existing in various normal states and excited states, char-
acterized by distinctive energy-values. The emission of alpha-
particles from radioactive nuclei has been studied with a new precision,
and leads to the same conclusion. The astonishing feats achieved
with bombarding particles of lesser energy have not lessened the hope
of achieving startling things with particles of greater. ■

Cockcroft and Walton, inspired by theory, had built an apparatus
for producing half-million-volt protons, and had proved them able to
transmute. The proton-streams had not, however, been greater than
five microamperes (one microampere or /za = 6.28-10'^ protons per
second). Next Oliphant and Rutherford, inspired by that result,

CONTEMPORARY ADVANCES IN PHYSICS 111

proceeded to build an apparatus in which the maximum voltage should
not go above a quarter of a million, but in recompense the stream of
protons should be raised to a hundred microamperes. Another
alteration: previously the stream had been a mixture of protons with
heavier ions and neutral particles — now Oliphant and Rutherford
introduced a magnetic field, adjustable and strong enough to bring
either the protons or the more massive ions separately against the
target. The magnetic field also assures that all the particles striking
the target shall have nearly the same speed, something not completely
guaranteed by the constancy of the voltage.

The scheme of this device is sketched in Fig. 2, where the course
of the proton-stream is traced (rather too pictorially, I fear!) in a
sweeping arc from its origin in the discharge-tube R, to the target T
where the element to be transmuted awaits the impacts. In the
discharge-tube all the parts are of steel, and the block C and cylinder
B conjointly form the cathode, while the oil-cooled block D and
cylinder A conjointly form the anode. This unusual material and
structure are required partly to minimize cathode-sputtering, and
partly to take care of the great amount of heat which is steadily
developed in the tube, inasmuch as for the best supply of protons a
voltage of 20,000 and a current of many milliamperes are demanded.
Something like a twentieth of the current in the discharge is borne
through the hole in the cathode by protons (and other positive ions
of greater mass, if such there be) ; and in the space between C and E
these particles receive from an electric field most of the kinetic energy
with which they strike the target. In this space and in the region
where the magnetic field comes into play, the density of the gas must
be kept extremely low, despite the fact that there is an open passage
into these spaces from the discharge-tube where the density must
always be great enough to sustain the discharge and the supply of
protons. This is a task for powerful pumps, which must be kept
continuously at work pumping away from the lower chambers the gas
which is steadily draining out of the discharge-tube through the hole
and must as steadily be replenished by feeding fresh hydrogen in from
above. It is no small part of the difficulty of the experiment, that
the discharge-tube and the source of its power and the source of its
hydrogen must all be maintained at scores or hundreds of thousands
of volts above the potential of the ground, in order that the observing-
apparatus may itself be at ground-potential. The transmutations are
observed by detecting the fragments which issue through the very
thin mica pane of the window W.

112

BELL SYSTEM TECHNICAL JOURNAL

Until the building of this apparatus proton streams had been so
scanty, that to bring about disintegrations in measurable number it
had been needful to project the protons against thick layers of dense
matter. In going through these layers they were slowed down and

V///////////

'ZZZZZZZZZZZZ2

Fig. 2-

-Apparatus of Oliphant and Rutherford for producing transmutation by
intense streams of protons. {Proceedings of the Royal Society)

stopped, and there was no direct way of telling whether the observed
transmutations were achieved by protons of full speed, or by those
which had already lost some energy, or both. Moreover if the energy
of the bombarding particles was raised, the number of disintegrations
infallibly went up, but a part of this increase (and sometimes the

CONTEMPORARY ADVANCES IN PHYSICS 113

whole of it) was certainly due to the fact that the faster particles
went farther into the layer and struck more nuclei. There is no need
to labor the point: it is obviously desirable to do the experiments with
a film so thin that each oncoming proton either strikes a nucleus with
its full and unabated initial energy, or else goes through the film and
away without any impact at all. This ideal was closely approached
by Oliphant and Rutherford, when they got countable numbers of
fragments from films of lithium and boron (deposited on blocks of
steel or iron) which were so thin as to be invisible, and of which the
latter was known to consist of only seven-tenths as many atoms as
would suffice to cover the iron surface with a single monatomic layer.
(The curves of Fig. 16 were obtained with these films.)

This is a success which proves it possible to investigate films con-
sisting each of only a single isotope of the element in question; for
feeble as are the ways of separating isotopes in all but a few very
favorable cases, they yet are powerful enough to produce pure mon-
atomic layers. This article will amply show how valuable will be the
privilege of getting data from a single isotope, of lithium or boron for
example; already there are several cases of important antagonistic
theories, the decisions between which will be given once and for all
by such data.

The apparatus devised by E. O. Lawrence and developed in his
school at Berkeley is of a singular ingenuity, inasmuch as in it ions
are accelerated until their energies are such as would be derived from
an unimpeded fall through a potential-difference of literally millions
of volts, and yet the greatest voltage-difference at any moment between
any two points of the apparatus is only a few thousands. It owes
its elegant compactness to the lucky fact that when a charged particle
is moving in a plane at right angles to a constant magnetic field, and
consequently is describing a succession of circles, the time which it
takes to describe a single circle is the same whatever its speed. One
sees this readily by writing down the familiar equation,

mv"l p — Ilevjc,

in which e, m, v stand for the charge (in electrostatic units), mass, and
speed of the ion and p for the radius of curvature of the circle, and
on the right we have the force exerted by the magnetic field H upon
the ion and on the left the so-called "centrifugal force" to which it is
equal. The radius p varies directly as v, but the time T = Irp/v
which the ion takes to describe a circle is independent of v. This is
no longer true if the ion is moving so fast that the foregoing classical
equation must be replaced by its relativistic analogue, but fortunately

114

BELL SYSTEM TECHNICAL JOURNAL

the desired results are attained without forcing the speed to such
heights.

Suppose now that while the ion is describing its consecutive circles
each in a time T, its speed is suddenly increased ; it continues to make
circles, of a larger radius but with the same duration. Suppose that
the increase occurs twice in each cycle, at intervals Tjl ; the path is a
succession of semicircles each broader than the one preceding but all
described in equal time. Now we arrive at Lawrence's device. The
ions circulate in a round flat metal box, sliced in two along one of its
diameters (Figs. 3, 4) ; and every time that one of them passes from

Fig. 3 — Diagram of the Lawrence apparatus for cumulative accelerations of protons
and other ions with auxiliary magnetic field. (After Henderson)

within half-box B to within half-box A it is accelerated by a voltage-
difference existing between B and A, and every time that it passes
from within A to within B it is again accelerated by a voltage-difference
between B and A. Of course if this voltage-difference remained the
same, the ion would lose at the latter passage just the energy which it
gained at the former; but here is precisely the distinctive feature of
the method : the potential-difference between the two half-boxes is reversed
in sign between each two consecutive passages. So rapidly do the
successive passages follow on one another, that if the intervals between
them were unequal it would probably be impossible to devise any
mechanism that would perform the potential-reversals at the proper

CONTEMPORARY ADVANCES IN PHYSICS

115

moments, but the felicitous law of the equality of the intervals makes
all easy — all that is needed is to connect an oscillator of the proper
frequency (determined by the strength of the magnetic field and the
charge and mass of the ions) across the pair of half-boxes. ^'^

Fig. 4 — Photograph of the apparatus sketched in Fig. 3. (E. O. Lawrence)

The sketch of Fig. 3 is of the apparatus wherewith Henderson
observed the transmutation of lithium by protons (pp. 140-142).
It is filled with hydrogen of a low density, so that electrons proceeding
from the hot filament F at the center ionize the gas and produce a
sufficient number of protons. These are whirled around and around
in ever-widening semicircles, till after a number of circuits which may
be as high as one hundred and fifty they arrive at the boundary of the

^^ One may do without the magnetic field, arranging to have the ions proceed along
a straight line and to accelerate them at definite points along that line, by voltages
produced in rhythm by an oscillator; the points of application of the voltages must be
spaced according to a particular way, and the apparatus is inconveniently long, being
longer the lighter the ion; it has been successfully employed with mercury ions by
Lawrence and some of his colleagues.

116

BELL SYSTEM TECHNICAL JOURNAL

half-box B opposite the charged electrode D, which deflects them
enough to bring them into the cup-shaped receptacle at the far end of
which the crystals of lithium fluoride are spread. The fragments of
lithium nuclei which are observed are those which escape to the left
in such directions as to enter the Geiger counter G. In this apparatus
the radius of the outermost circle was 11.5 cm., the magnetic field
14,000 gauss; the potential-difference between the half-boxes never
attained as much as 5000 volts, but it was reversed 4.2 millions of

Fig. 5 — The Lawrence apparatus for cumulative acceleration, beside the colossal
magnet between the poles of which it is placed. (Lawrence)

times in a second, so that after three hundred reversals the protons
had arrived at the limit of the box and were ready to strike the lithium
nuclei with an energy of 1.23 MEV. By using a bigger pair of half-
boxes in a more extensive magnetic field, this energy could be aug-
mented; and by viewing the size of the magnet in Fig. 5 one sees
what an augmentation is now imminent. The currents were inferior
to those achieved by the apparatus of the Cavendish school, being
mostly of the order of a few millimicroamperes (one millimicroampere
or mixa = 10~* na).

CONTEMPORARY ADVANCES IN PHYSICS 117

Detection and Measurement of Transmutation

While thus the scope of transmutation has been so vastly extended
in the past eighteen months, there is one limit which has not yet been
passed. No product of transmutation has yet been detected by any
chemical means. Many a plate of metal has been bombarded with
protons or with alpha-particles, but no man has seen it change into a
plate of another metal, nor alter in any of its chemical properties;
many a tubeful of gas has been bombarded, but no man has observed
the qualities or the spectrum-lines of another gas appearing in the
content of the tube. All that is ever observed is an outpouring of
material particles from the piece of bombarded matter; particles of
such a nature, that they must come from the nuclei of the atoms.
One expects this statement to go out of date from one morning to
the next; but at the moment of this writing it is still as true as it
was in 1919 when Rutherford first disintegrated nuclei, and broader
in one respect only. From 1919 until 1932, one would have said
"charged particles"; but since the winter of 1932, it is known that
either charged particles or uncharged may be driven out of nuclei,
by the appropriate impacts.

Thus there are two great experimental problems, and not one only;
beside the problem of producing the streams of bombarding corpuscles,
there is that of detecting and of recognizing the particles which fly
forth from the bombarded nuclei — the "fragments," I will say. There
is a grave objection to this term, and to the common name "disintegra-
tion" for the process. Both suggest a picture of the nucleus as a
structure of pre-existing pieces which the impact breaks apart and
scatters. This picture is surely incorrect, for there are cases in which
the fragments contain the susbtance of the impinging corpuscles. In
fact, if we define "fragment" — as we should — to include the part which
in most of the experiments does not escape from the target bulk, we
may say that this kind of case is frequent, and perhaps indeed that
there is no other kind ! Nevertheless we seem to be unable to get along
without the words "disintegration" and "fragment."

For detecting protons and more massive fragments which are
charged, there are three methods.

Tho: first method (A) is that of observing the scintillations, which fast
charged particles produce when they impinge on fluorescent screens.
This is the classic and historic method, by which were made the
earliest proofs of transmutation by impact of alpha-particles (which
I described at length in the earlier article) and also the earliest proof
of transmutation by protons. Of late years this method has been
largely displaced by the others. Few people outside of the Cavendish

118 BELL SYSTEM TECHNICAL JOURNAL

Laboratory and the Institut fur Radiumforschung in Vienna have
ever submitted themselves to the long, tedious and nerve-racking
process of counting thousands of dim flashes for periods of hours in
darkened rooms with dark-adapted eyes; and if two disagreed as to
what was observed, there was no objective way of deciding between
them. The newer methods abolish this strain; they can readily be so
shaped as to leave a permanent record, which anyone may consult and
analyze for himself; and they are capable of measuring the ionizing
power of the fragments. Nevertheless the eldest method still retains
the unique advantage that no barrier whatever, not even a gas, need
intervene between the detecting screen and the source of the fragments;
and also it is often employed by those accustomed to scintillations as
a check upon the others.

The second method (B) is that of the expansion-chamber or cloud-
chamber of C. T. R. Wilson, whereby the tracks of ionizing particles
across a gas are made visible by droplets of water which condense
upon the ions. This is the splendid invention which is the joy of all
who write or lecture on atomic physics, since it enables them to deco-
rate their exposition with pictures which make real the things of
which they speak. It has virtue for the investigator also, especially
since it may show in a single vivid photograph how many fragments
there are formed in a single process, what are the directions in which
they fly away, and how far they are able to travel through the gas.
The curvature of the track in an applied magnetic field supplies the
value of the momentum of the particle which made the track, if the
nature of the particle be known; and this last may often be guessed
from the aspect of the track, or assured by independent data. The
major disadvantage of the method is, that the apparatus records only
the particles which fly off during about a hundredth of a second, and
then lies idle for several seconds or even minutes while it is being
prepared for its next brief interval of effectiveness.

The third method (C) — or group of methods rather, for the variants
are legion — is the detection by purely electrical methods of the ions
which the fragments produce as they shoot across the gas of an
ionization-chamber. A fast-flying charged particle loses on the aver-
age 30 to 35 electron-volts for every ion, or rather every ion-pair,
which it produces.'^ To see the utility of this theorem, turn it around ;
the number of ion-pairs produced by a fast charged particle going
through a gas is about a thirtieth of the number of electron volts
which it loses in its transit. A fast alpha-particle, such as are spon-
taneously emitted by radon, or constitute the fragments springing out

'' "Electrical Phenomena in Gases," pp. 52, 70-71.

CONTEMPORARY ADVANCES IN PHYSICS 119

of lithium bombarded by protons, has about eight milhon electron-
volts; if it enters an ionization-chamber filled with gas so dense that
it is brought completely to a stop, the ion-pairs appearing are about
a quarter of a million. The upper limit occurring in practice is
possibly twice as high, but is very rarely met with; there is no lower
limit, but every incentive to push downward and ever downward the
least amount of ionization which can be detected.

Twenty-five years ago, it would have been impossible to detect by
electrical means so few as a quarter of a million ions. (The total
number produced e.g. by an alpha-particle was determined by meas-
uring the total ionization produced by a known and very great number
of particles.) This problem was however destined to be solved in
many ways, which I will group under four headings :

(CI) By arranging to have each particle touch off a brief but violent
discharge, something like an invisible spark, in the gas of the ionization-
chamber. There is a strong electric field applied between the elec-
trodes of the chamber, whereby the "primary" ions which the particle
forms as it travels across the gas are caused to produce (directly and
indirectly) vast numbers of extra or "secondary" ions; and these
suffice to make a sensible effect in the external circuit. The idea was
first put into practice by Rutherford and Geiger in 1908, and the
scheme is commonly known by Geiger's name. One of the electrodes
must be either a fairly sharp point or a fairly thin wire, and there are
a number of empirical rules (some partially understood, some not at
all) about the size and shape of the chamber, the proportioning and
the conditioning of the electrodes, the nature and the purity and the
density of the gas, and the magnitude of the field. The voltage
across the gas must lie within a definite range, often pretty narrow;
if it is lower the particles do not produce discharges, if it is higher a
single discharge may last indefinitely. The ratio of the number of
secondary to the number of primary ions is usually not constant and
usually not measured; most of the various forms of the device serve
solely to detect or count the particles, and they are known as "Geiger
counters." Often a loudspeaker is connected into the circuit of the
ionization-chamber, and each discharge produces an audible clack, so
that by the Geiger method one hears the passage of a corpuscle as by
the Wilson method one sees it. Sometimes the discharges are recorded
and the record examined at leisure.

(C2) By modifying the foregoing scheme so that the number of
secondary ions shall be proportional to the number of primary ions,
and a measurement of their total charge shall give at least a relative
value of the ionizing-power of the traversing particle. This is a

120 BELL SYSTEM TECHNICAL JOURNAL

recent achievement of Geiger and Klemperer. The process may be
called internal amplification of the primary ionization, the amplification
being in a constant proportion, or, as people carelessly call it, "linear."

(C3) By developing an electrometer or electroscope so sensitive that
it is able to detect and even measure the total charge of a few thousands
of ions, without amplification. This was first achieved, or at any rate
applied to transmutation, by G. Hoffmann of Halle, and his associate
Pose; the latter was able to observe fragments of aluminium nuclei
(ejected by alpha-particles) which produced as few as three thousand
ion-pairs. The major difficulty seems to be, that the electroscope
takes a large fraction of a minute to perform its deflection and then
recover its readiness to respond to another particle. Pose in his
experiments observed only some thirty fragments to the hour.^^

(C4) By applying external amplification to the feeble impulse which
the primary ions due to a single particle produce in the external
circuit, and which is imperceptible to an electroscope of normal and
convenient quickness of response. This is done by developing the
superb techniques of amplification which modern vacuum-tubes have
rendered feasible, and like the three foregoing schemes is an achieve-
ment of the last few years, having been carried on especially by Wynn-
Williams of the Cavendish Laboratory and Dunning of Columbia.

I show as Fig. 6 three records made with Dunning's apparatus,
wherein every vertical line is due to an ionizing particle, and is pro-
portional in length to the number of ions which the particle produced
in crossing a shallow chamber. ^^ The lines of great and nearly uniform
length which appear in record (a) are due to alpha-particles from
polonium; these all had nearly the same speed and were moving in
nearly parallel lines when they entered the chamber, and it is evident
that in crossing the gas they all made nearly the same amount of
ionization; they left with a good deal of their initial kinetic energy
unspent. The lines in record {b) are caused by protons; their diversity
in length is chiefly due to the wide variety of speeds which the protons
had when they entered the chamber, for these were fragments of the
disintegration of aluminium by alpha-particles, and therefore had a
broad distribution-in-speed (page 147). As these words imply, and
as I will stress presently, the ionization produced by a charged particle

^^ It deserves to be recorded that in their blank experiments, Hoffmann and Pose
during one research observed deflections at the average rate of 1.22 per hour, but
observed altogether 197 of them !

" I am much indebted to Dr. Dunning for these pictures, made especially for this
article. He writes of {b): "The minimum amount of ionization detectable here is
well under 1000 ions; probably it could be pushed down to 250 ions." Consecutive
dots at the bottom of each record mark off the minutes.

CONTEMPORARY ADVANCES IN PHYSICS

121

of given kind depends on its speed; the greatest amount which (in
this particular chamber) a proton could ever produce, with its most
favorable speed, is indicated by the longest lines in {h), and one sees
that even these are definitely shorter than the lines in {a) due to

^-- f'-^-'Tf^-fr'

(a)

1 — ii — -f^i ■ ii--M-J — hH :■

{b)

— .j . — f — J i —

•i

\i \

r- ■■■■■"

—

1! 1

- t

[

r t f

m ' •

ic)

Fig. 6 — Three records of the ionization produced by individual particles in a
shallow ionization-chamber: (a) alpha-particles of nearly the same speed, (b) protons
of various speeds, (c) particles of several kinds which had been set into motion by
impacts of neutrons. The fogging along the base-lines is much fainter in the original
records than in these reproductions. (J. R. Dunning)

122 BELL SYSTEM TECHNICAL JOURNAL

alpha-particles.^^ The lines in record {c) were obtained when neutrons
were traversing the chamber and a piece of paraffin outside of it; not
they, but the charged nuclei which they strike and impel, are producing
the record. Those lines which are longer than any in {h) are certainly
not due to protons; they must be caused by recoiling nuclei of the
atoms of the gas which fills the chamber (air), and which have various
speeds because the neutrons strike them more or less glancingly (and
probably do not themselves all have the same speed). The shorter
lines are due in part to such nuclei, chiefly to protons ejected from the
paraffin in such directions that they cross the chamber. Some are
very short indeed, half-lost in the dusky haze due to the perpetual
wiggling of the oscillograph mirror caused by gamma-rays; they are
made by the fastest of the protons. Observations by expansion-
chambers and with applied magnetic fields have proved this classifica-
tion of the particles.

All of these methods are available for detecting charged particles
which are protons or alpha-particles or corpuscles of a yet greater
mass than these. For electrons the problem is harder.

An electron of given energy — say x thousands of electron-volts — is
able to make roughly as many ion-pairs in a gas as could a proton or an
alpha-particle of equal energy: that is to say, about 30.Y. Nevertheless
it produces much less ionization in an ordinary chamber than either
of these" last. This seeming paradox is due to the facts that the ion-
pairs produced by the electron are relatively far apart and the loss
of energy per centimeter of path is correspondingly low, so that in an
ionization-chamber of reasonable dimensions and customary density
of gas the traversing electron produces only a few hundreds or perhaps
one or two thousands of ion-pairs before it reaches the opposite side
of the chamber and plunges into the wall.

This is made evident by the Wilson method, the tracks of electrons
appearing much thinner — less richly peopled with droplets, that is to
say — than those of alpha-particles or protons. The expansion-
chamber therefore is available for observing fast electrons, and so to a
certain extent is the Geiger counter, which skilful observers can adjust
so that it will react to these bodies. None of the other methods has
yet been used with success. The ions produced by a single electron
in an ionization-chamber are apparently too few to observe w^ithout
amplification or even to amplify successfully, and the scintillations
too faint. If one has neither expansion-chamber nor Geiger counter
available, the only thing to be done is to measure the total ionization

''' The contrast is much more striking than the records suggest, for. the amplifica-
tion was fourfold greater when (b) and (c) were made than when (a) was made.

CONTEMPORARY ADVANCES IN PHYSICS 123

produced by great numbers of electrons, and attempt to estimate these
numbers. This is done in the study of the beta-rays or fast electrons
emitted from radioactive nuclei, and in the study of cosmic rays; but
the method has not yet been applied to the rays emitted from atoms
undergoing transmutation by impacts, and apart from Joliot's observa-
tions on positive electrons (page 102 supra) nothing yet is known of
any electrons which may be emitted by these.

Since individual electrons are so difficult or impossible to observe
by the customary methods, one might suppose that at any rate they
never annoy the observer. This unluckily is not so; for if electrons
are numerous, they may keep the electrometer needle (in the method
C4, for example) in a perpetual tremor, producing a so-called "back-
ground" over which even the strong sharp impulses due to alpha-
particles or protons may fail to stand out. It is even possible for a
chance coincidence or near-coincidence of several electrons to make a
record which cannot be disinguished from that of a single particle of
greater ionizing power. The scintillation-method suffers from a like
defect, for if the fluorescent screen is heavily bombarded with electrons
— or with gamma-rays, which liberate electrons from the fluorescent
stuff and the surrounding matter — it shines all over with a feeble
glow, against which the flashes made by more massive ions are difficult
to discern. The most casual student of transmutation cannot fail to
notice that polonium is generally used, of recent years, as the source
of alpha-particles for bombardment. Probably he infers that either
it is especially abundant or else supplies especially fast particles.
But in both respects polonium is inferior to another customary source,
radon mixed with its descendants radium A and radium B. It is
used because it emits no gamma-rays but feeble ones of low penetration,
whereas the other source pours out abundant and powerful photons
which flood any nearby ionization-chamber with electrons and confuse
the electrometer. Dunning's amplifying circuit, whereby he detected
charged nuclei set into motion by neutrons, was so devised as to
discriminate against the feeble but many impulses produced by these
electrons and in favor of the occasional stronger ones produced by the
massive particles; and this device enabled him to use, a source of the
latter type providing fifty times as many alpha-particles to engender
the neutrons, as the largest amount of polonium ever employed.

Neutrons, I recall, are detected by observing the protons and more
massive nuclei which they convert by impact into fast-flying ionizing
particles, and photons by observing the electrons on which they have
the like effect; the problems of getting the data are thus not new,
it is the problem of interpreting them which is changed.

124

BELL SYSTEM TECHNICAL JOURNAL

The next important question is, how the fragments are identified as
protons, or as alpha-particles, or otherwise, from the data. Few as
yet are the cases in which the identification is full and undeniable.
In the earlier paper I described Stetter's measurements of charge-
to-mass ratio for the fragments produced by impacts of alpha-particles
against boron, carbon, fluorine and aluminium, which gave values
identical with that for protons within the observational uncertainty
of five per cent. As for the fragments produced by impacts of protons,
the best direct evidence is that which appears in Fig. 7. Cockcroft

? 5

z
o
\-

111 ^

9 2

o I

Hill

5.0 5.6 6.0 6.5 7.0

AIR EQUIVALENT IN CENTIMETERS

Fig. 7 — Ionization produced in a shallow chamber by fragments (of the trans-
mutation of lithium by protons) which have passed through screens of various
thicknesses. (Cockcroft and Walton)

and Walton had an ionization-chamber only 3 mm. across, and the
fragments from bombarded lithium traversed it completely, producing
a few thousand ion-pairs apiece which were detected and measured
with the aid of an amplifying circuit of Wynn-Williams according to
the method C4. When mica sheets were interposed in the path of the
fragments from the lithium, they were slowed down but still kept
energy enough (so long as the sheets were not too thick altogether)
to travel across the chamber; and the curve of Fig. 7 represents the
number of ion-pairs produced per fragment, as function of a quantity
X proportional to the thickness of mica which the fragments have

CONTEMPORARY ADVANCES IN PHYSICS 125

traversed ("air-equivalent" of the mica, p. 127 infra) }^ The point
is, that exactly the same curve was obtained when a beam of alpha-
particles was projected through the same thicknesses of mica into the
same chamber. Mere similarity in the shape of the curves would
prove nothing, for this is the shape obtained with all kinds of charged
particles, electrons and protons and more massive charged nuclei; in
particular, every such curve rises from zero to a maximum and there-
after descends continually as the energy of the particles is raised
indefinitely upward from the least value sufficient for ionization. ^^
However, the ordinates of the curve of Fig. 7 are equal to those of the
alpha-particle curve, and about four times as great as would have been
observed with protons; and this it is which proves the fragments to
be alpha-particles. Almost as good a proof could be made by two
measurements: by measuring the range of the fragments and the total
ionization produced by any fragment in a chamber deep enough to
swallow it up, and comparing the latter datum with the ionization
produced in the same chamber by an alpha-particle of equal range.
This proof, or some other substantially like it, has been adduced in
certain cases. When alpha-particles are the agents of the transmuta-
tion, the same test has proved in several cases that the fragments are
protons. In some cases the test has not yet been applied.

I have already had to speak of interposing mica in the path of the
fragments, in order to learn something about them. This is a pro-
cedure with which it is necessary to be familiar. It would be very
pleasant indeed to be able to apply electric and magnetic deflecting
fields to a narrow stream of fragments all flying in the same direction,
for one could then spread it out into a velocity-spectrum, and not
only identify the corpuscles perfectly but also determine their distribu-
tion-in-range, which as we shall presently see is of the first importance.
This has not yet been done, partly (I presume) because of the high
fieldstrengths that would be needed, chiefly because the available
streams of particles are too scanty. It will be a happy day when at
last we get streams of fragments so intense that they can be dispersed
into a velocity-spectrum which will appear imprinted on a photographic
film, as has been feasible for years with beta-rays. For the time being
we must be content with curves such as many figures in this article
display, Figs. 8 and 9 and 1 1 for example.

^^ The quantity plotted as ordinate is obtained from such records as those of Fig. 6,
in which every fragment produces a vertical line. Cockcroft and Walton observed
many such lines for each thickness of mica, and ascertained in each case the most
frequently-occurring value of line-length.

^^ "Electrical Phenomena in Gases," pp. 40-44, 70-71. Such a curve as that of
fig. 7 is sometimes called a "Bragg curve."

126

BELL SYSTEM TECHNICAL JOURNAL

These are curves in which the abscissa stands for the thickness of a
special kind of matter (air of a standard density) interposed in the
path of the fragments, and the ordinate for the number of fragments
detected on the far side of that matter; I will call them "integral
distribution-in-range" curves representing the number /(x) of particles
able to traverse thickness x. Were one to differentiate them, one
would get the "differential distribution-in-range" curves, representing
a function f'{x) such that f'{x)dx stands for the number of particles
able to traverse thickness x but not additional thickness dx — the
particles which are said to have "ranges" between x and x + dx.

» ■* * 4 « " n ■ ii r^ ^

I I I 1 1 1 •*-

1.5 2 3 4 5

AIR EQUIVALENT IN CENTIMETERS

Fig. 8 — Integral distribution-in-range curve of the fragments resulting from bombard-
ment of lithium by protons. (Oiiphant Kinsey & Rutherford)

These, however, are usually not plotted,''^ and one must accustom
himself to draw the proper inferences from the integral curves.

The clearest of these to read are those which are shaped like a
staircase, with steep rises connecting horizontal parts called paliers or
plateaux. A steep rise extending over a narrow interval of x signifies
a "group" of fragments all having ranges close together. A plateau
extending over a broad interval of x signifies that no particle has a
range comprised anywhere in this interval. An integral curve in the
form of a staircase therefore implies the analogue of a line-spectrum,

" One of the rare examples is reproduced in "Transmutation," B. S. T. J., \'o\. X,
p. 650 (Oct. 1931), from the work of Bothe and P>anz.

CONTEMPORARY ADVANCES IN PHYSICS

127

the particles being classifiable into groups each with its characteristic
speed. But if in such a curve there is a long sloping arc (as in Fig. 9),
it implies the analogue of a continuous spectrum, there being particles
of all ranges over a notable interval.

The "stopping" or "absorbing" screens which are used in deter-
mining these curves are usually sheets of mica or of aluminium.
The curves are not however plotted against the actual thickness of
the interposed strata of mica or whatever else the substance may be,
but against the "air-equivalent" or thickness of the stratum of air of
standard density ^^ which is known by separate experiments to have
the same effect in slowing down and stopping charged particles, the

-5

•

DEUTONS-^

6-«-PR0T0NS

_ . -

^-^

^ "N

JO 250
)(r

^5 200

Oct 150

(Tq.

0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15

AIR EQUIVALENT IN CENTIMETERS

Fig. 9 — Integral distribution-in-range curve of the fragments resulting from bombard-
ment of lithium by deutons. (Oliphant Kinsey & Rutherford)

same "stopping-power." It is the air-equivalent which is the quantity
X of the preceding paragraphs and the abscissa (often termed "ab-
sorption") of Fig. 8 and nearly all other such figures. The ratio
between the actual thickness of a layer of matter and the equivalent
thickness of air is roughly (but only roughly) the reciprocal of the
ratio of their densities. The sheets of metal or of mica used in the
experiments are therefore very thin (it has been possible to make
screens of mica so tenuous that their air-equivalent is only 0.15 mm.)
and the thinnest must be bolstered up by stiff metal grids, of which
the wires block a considerable fraction of the beam. It is also possible

'^ There are unluckily two standards of density, one being that of air at 0° C.
and 760 mm. Hg, the other that of air at 15° C. and 760 mm. Hg; sec "Transmu-
tation," footnote on p. 643, B. S. T. J., Oct. 1931. The latter is used in this article.

128

BELL SYSTEM TECHNICAL JOURNAL

to use air (or some other gas) of adjustable density; when the scintilla-
tion-method is employed, the gas may fill the entire space between
the source of the fragments and the fluorescent screen; with other
methods of detecting the fragments, it must be contained in a cell
which the stream enters and leaves through windows of mica or
similar substance.

G 4.5

i 3.5

i'3.0

' / I

\ • / 1

• ^ ,< I

• 1

-— ^~i ■! —*

0.5 0.6 0.7 0.8

0.9 1.0 I.I 1.2 1.3 1.4 1.5 1.6

AIR EQUIVALENT IN CENTIMETERS

Fig. 10 — Ionization produced in a shallow chamber by the least penetrating
fragments from the transmutation of lithium by protons. (Oliphant Kinsey & '
Rutherford)

There is an interesting and important way of confirming the steps
in an integral curve such as those of Figs. 8 and 9. Near the rise of ^
such a step, the thickness of the intercepting matter is such that many
particles are approaching the ends of their ranges when they emerge]
from the last of the screens. Suppose that this last screen is adjoined
by a very thin ionization-chamber, like that with which the curve

CONTEMPORARY ADVANCES IN PHYSICS

129

of Fig. 9 was obtained. Let the air-equivalent x of the total thickness
of the screens be varied, and let the average number of ions produced
per particle in the chamber be measured and plotted as function of x.
Recalling Fig. 7 and what was said in respect to it, the reader will see
that the resulting curve should have a peak wherever the integral
distribution-in-range curve has a step. This has been verified several
times, and there are cases in which these peaks have been taken as

uJ 50
ffl

END

1.5 2.0 2.5 3.0 3.5

AIR EQUIVALENT IN CENTIMETERS

Fig. 11 — Integral distribution-in-range curve of the fragments resulting from the
bombardment of boron by protons. (Oliphant & Rutherford)

clearer evidence for the existence of groups than the shape of the
integral curve itself (Fig. 10). Peaks may also appear in a curve of
which the ordinate is the total ionization produced in the very thin
chamber by all the fragments which enter it.

Anyone at all acquainted with physical experiments will readily
suspect that the steps of actual integral cuives "ought to be" steeper
than they are. I mean : that he will form the hypothesis that perpen-

130 BELL SYSTEM TECHNICAL JOURNAL

dicular rises would be observed instead of rounded-off and sloping ones,
if only the pencil of fragments passing through the absorbers were
ideally narrow and cylindrical, and were produced by bombardment
of atoms with particles all of the same speed; and he will attribute the
rounding-ofif of the steps to the facts that the fragments actually form
a divergent and conical beam, and the atoms from which they come
have been struck by impinging particles of diverse speeds. This idea
is strongly supported by the facts that the steps are notably steepened
when the divergence or "aperture" of the beam of fragments is
reduced, and when the diversity of speeds among the bombarding
particles is narrowed.

The former of these variables is controlled by the slits and dia-
phragms which bound the beam, and the latter by the thickness of
the bombarded target whence the fragments proceed; for the bom-
barding particles are slowed down as they dive deeper into the target,
and nuclei at different depths receive impacts of different energy, and
thus there is a wider diversity of speeds among the particles when they
finally make their impacts than there is among them when they start
from their source. But as one cuts down either the thickness of the
target or the aperture of the beam of fragments, one reduces the
number of fragments which come to the detecting apparatus, and
reaches a limit when this number becomes too small to be observed
in any convenient time. Progress in approaching ideal conditions
therefore depends on progress in multiplying the number of fragments
by multiplying the strength of the bombarding beam. We may count
on a yet greater steepening of such steps as those of Fig. 8, when the
enormous streams of bombarding protons produced by Oliphant and
Rutherford are applied to very thin films and the distribution-in-range
of the resulting fragments is measured. A corresponding improvement
of the curves obtained when alpha-particles are the bombarders is
still in the not-immediate future. Whether under ideal conditions
the steps would be absolutely perpendicular, and all the fragments of
a group have exactly the same speeds, is not as yet to be safely inferred
from the data.

There remains the great problem of converting distribution-in-range
curves into distribution-in-speed or distribution-in-energy curves, and
thus determining the energy or the speed of fragments belonging to
a group of which the range is known. The recent developments of
research in transmutation and in cosmic rays have elevated this to
the rank of the major problems of physics. For alpha-particles of
ranges of 8.6 crn. and less, it is practically solved by empirical means;
for such alpha-particles are supplied in such abundance by radioactive

CONTEMPORARY ADVANCES IN PHYSICS 131

bodies that it has already been feasible to measure by deflection-
methods the speeds corresponding to a large number of different ranges,
and plot an empirical speed-i'5-range curve which is fixed by so many
points of observation that there is no important uncertainty in making
interpolation between these. For alpha-particles of range superior to
8.6 cm., such as often occur among fragments of transmutation, it has
heretofore been necessary to extrapolate ; but very lately the empirical
curve has been extended onward to 11.6 cm., thanks to a powerful new
magnet at the Cavendish which is able to deflect the paths of alpha-
particles of even such rapidity.^^ With protons our knowledge of the
range-V5-energy relation is less extensive and less accurate, and an im-
provement thereof should be one of the first and most important by-
products of the new methods for imparting high energies to ions. For
charged nuclei of other elements than hydrogen and helium, relatively
little is assured (what is known has been found out chiefly by Blackett
and his school)-" ; but this lack has not as yet been much of an im-
pediment to the study of transmutation, except in certain cases involv-
ing impacts by neutrons.

Transmutation by Impacts of Protons and Deutons

The earliest element to be transmuted by protons in the laboratory
— indeed the first to be transmuted by man with any agent other
than the alpha-particle — was lithium. It was fortunate that Cock-
croft and Walton began with this element, for its behavior turned out
to be uniquely lucid. In most disintegrations, a single fragment is
detected, and there must be a massive residue which remains unseen,
staying hid within the substance of the bombarded target. But in
some at least of the transformations which occur when lithium nuclei
are struck by protons or deutons, there seems to be no hidden residue;
every fragment is observed and recognized. These are processes of
"nuclear chemistry" of which we fully discern both the beginning
and the end; and they are described by the quasi-chemical equations:

'^Rutherford et al., Proc. Roy. Soc. 139, 617-637 (1933). The empirical curve
departs slightly from a third-power law (range proportional to cube of speed) and
the results are expressed by an empirical formula for the departure. See also G. H.
Briggs, Proc. Roy. Soc. 139, 638-659 (1933).

2° See N. Feather, Proc. Roy. Soc. 141, 204 (1933) and literature there cited.
The observations are made upon tracks which appear in Wilson chambers when the
contained gas is bombarded by alpha-particles, and which are the tracks of objects
of atomic mass that have suffered violent impacts. It is presumed (though not
always proved) that these objects are solitary or "bare" nuclei, not accompanied
by any of the orbital electrons which attended them before the impacts. Some (but
not all) of the data conform to the empirical rule that the ratio of the ranges of two
nuclei of masses nti and m-. and of charges Z\e and Z>e, when the two have the same
speed, is (wi/w2)(Zi/Z2)"-.

132 BELL SYSTEM TECHNICAL JOURNAL

,W + zW +To = 22He^ + Ti, (1)

iH2 + sLis + To = 22He^ + Tr, (2)

of which the first has already appeared in Part I. of this article.

These are to be regarded as equations for mass and energy, owing
to the equivalence of these two entities. Attached to the symbol of
each atom are its mass-number as superscript and its atomic number
as subscript (and, incidentally, every such equation must balance
when considered as an ordinary equation in either the mass-numbers
or the atomic numbers). The symbols To and Ti stand for the total
kinetic energy of the particles before and the particles after the trans-
mutation, expressed in mass-units. (I recall from Part I. that a
mass-unit is one-sixteenth the mass of an sO^^ atom, and that one
million electron-volts is equal to 0.00107 of one mass-unit.) The other
symbols then stand for the rest-masses of the nuclei of the atoms in
question. It would be proper, and in accordance with the spirit of
relativity, to leave out the symbols T^o and Ti and consider each of the
other symbols as standing for the total mass of the nucleus, viz. the
sum of its rest-mass and the extra mass resulting from its speed.
When hereinafter the symbols Tq and Ti are absent from such an
equation, the others are thus to be interpreted.

The suggestion thus is, that when a proton meets with a sLV nucleus
or a deuton with a sLi^ nucleus, either process ends in the formation
of two helium nuclei — alpha-particles — out of the substance of the
original bodies. It is further suggested that these nuclei share kinetic
energy amounting to Ti; and if they are emitted in directions making
equal angles with that of the impinging particles — the "symmetrical
case" which (as we shall see) is most commonly observed — they must
share Ti equally in order to assure conservation of momentum. Now
the rest-masses of all the nuclei figuring in equations (1) and (2) are
accurately known through the work of Aston and of Bainbiidge.
Taking them from Table I and substituting them into the equations,
and using the electron-volt for our unit, we get :

Ti = To -f 16.8- 10«, ■ (3)

Ti = To + 22.2- 10«, (4)

in the two cases, ^' and therefore expect alpha-particles paired with
one another, their kinetic energies amounting altogether to these values.

^^ For these numerical values and their uncertainties, see K. T. Bainbridge, Phys.
Rev. (2). 44, 123 (July 15, 1933).

CONTEMPORARY ADVANCES IN PHYSICS 133

It is the verification of these predictions which gives us such great
confidence that we have recognized the processes which really happen.

I have already said how Cockcroft and Walton proved that the
fragments, when lithium is bombarded by protons, are alpha-particles.
The integral distribution-in-range curve of these fragments, obtained
by Oliphant Kinsey and Rutherford with the apparatus of Fig. 2 and
proton-currents running up to SOixa, appears in Fig. 8; a.nd that for
the fragments created when deutons are used instead of protons
appears in Fig. 9. In both of these one cannot but be struck by the
beautiful long horizontal plateaux, and the sharpness of the steps
which end them on the right. The groups of fragments of which
these steps are the signs have ranges stated by the observers as 8.4 and
13.2 cm respectively, with uncertainties of ± 0.2 cm. (These figures
are evidently taken from the bottom of the step, probably because it
is assumed that under ideal conditions of narrow beam and thin
bombarded film — the actual beam had a divergence of about 15° and
the actual target was thick — the step would rise vertically from the
point whence it actually begins to rise obliquely.) The corresponding
energy-values are estimated as 8.6 and 11.5 MEV (millions of electron-
volts) respectively; and as Tq, the energy of the impinging protons,
is at most two-tenths of a million, these values may be compared
directly with the halves of the numbers in equations (3) and (4).
Meanwhile at Berkeley, Lewis Livingston and Lawrence were driving
deutons with an energy of 1.33 MEV— no longer negligible — against
lithium, and observing fragments with a range of 14.8 cm., corre-
sponding to an energy of 12.5 MEV; and this is to be compared with
half of 23.7 millions on the right-hand side of equation (4).

The agreement in the case of protons impinging on lithium is
admirable, and well within the uncertainty of the data. The agree-
ments in the cases of deutons impinging on lithium are ostensibly
not so good, but this is not so serious as it seems at first glance, because
of the required extrapolation of the range-2^5-energy curve of alpha-
particles (page 131), and because it is not always the "symmetrical
case" which occurs. For the present there is no compelling reason to
suppose that equation (2) is contradicted by the data.

A further point susceptible of test: if the processes described by
equations (1) and (2) are actual, the alpha-particles of the stated
ranges must be shot ofif in pairs, the two members of each pair flying
off in almost opposite directions — in directions which would be exactly
opposite were it not for the original momentum of the proton, but
which because of that momentum must make with one another an
angle slightly (and calculably) less than 180°. Cockcroft and Walton

134

BELL SYSTEM TECHNICAL JOURNAL

made the test with a pair of Geiger counters set on opposite sides of
the bombarded lithium, and got a positive result; but it is the ex-
pansion-chamber which is suited by its nature for supplying the most
magnificent of proofs. To achieve this, one must put the bombarded
target of lithium in the middle of the chamber, and photograph the
tracks from above; and since the bombarding stream must come
through vacuum while the chamber must be filled with moistened air,
the target must be separated from the air by walls of mica thick

FAST I
PROTONS

-SHUTTER

KWWWWWW^ kWWNWWWSM

MERCURY
VAPOR
LAMP

MICA WINDOWS
' SUPPORTED
ON GRID

Fig. 12 — Diagram of arrangement for observing tracks of fragments by the expansion-
method. (After Dee and Walton)

enough to withstand the pressure and thin enough to let the fragments
pass. The scheme is clearly depicted in Fig. 12. One notices that the
design is such that the pairs which are observed are those of w^hich the
directions are nearly at right angles to the proton-beam — the "sym-
metrical case" aforesaid.

This experiment was first performed by Kirchner of Munich, who
got several pictures of paired fragments from lithium bombarded by
protons. Fig. 13 shows an example. (The third track is rather

CONTEMPORARY ADVANCES IN PHYSICS 135

annoying, but it was quite an achievement so to adjust the conditions
as to get so few as three.) Many splendid examples have lately been
published by Dee and Walton of the Cavendish, and Fig. 14 is out-
standing among them because the bombarding stream was a mixture
of protons and deutons, and the picture shows two pairs of fragments,
one apparently due to each of the processes which I have been de-
scribing. Those of the pair marked &1&2 have the range of 8.4 cm.
agreeing with equation (1), while those marked aia2 go definitely
farther and even escape from the chamber, which makes it impossible
to measure their ranges. Dee and Walton therefore made the walls
of the target-capsule thicker, so that more of the energy of the frag-

Fig. 13 — Tracks of paired fragments, He nuclei resulting from impact of a proton on a
Li' nucleus. (Kirchner; Bdyrische Akademie)

ments should be consumed in them; the pairs which were obtained
with bombarding deutons now ended in the chamber and in the field
of view, and their ranges agreed with the 13.2 cm. obtained from the
curve of Fig. 9. At least two more of these pairs appear in Fig. 15.
Verification of a theory could scarcely go further or be more vivid !
Yet there is the additional point, that Kirchner found the angle
between the paired paths in his pictures to differ from 180° by just
about the amount required by the momentum of the proton.

However not every fragment observed when lithium is bombarded,
either by protons or by deutons, results from these superbly simple
interactions. Notice in Fig. 8 the two very much rounded steps,
suggesting groups of short ranges (1.15 cm. and 0.65 cm.); these are
confirmed by the maxima in the curve of Fig. 10 which has already

136

BELL SYSTEM TECHNICAL JOURNAL

been explained (page 129). Only tentative theories of these have
been made, and it would be of little use to expound them here.^^
Notice then in Fig. 9 the beautiful long sloping line adjoining the
plateau, and implying a continuous distribution over a wide interval
of ranges extending up to 7.8 cm. The numerous shorter tracks of
Fig. 15 are due to particles belonging to this continuum. Observe
last the integral distribution-in-range curve for the fragments from

Fig. 14 — Tracks of paired fragments, He nuclei believed to result from impact of a
proton on a Li^ nucleus and from impact of a deuton on a Li^ nucleus. (Dee and
Walton ; Proceedings of the Royal Society)

boron bombarded by protons. Fig. 11; notice that it displays no
definite step, but consists of a single sloping arc implying a continuum
extending to an upper limit, which on a magnified curve is found to be
at 4.7 cm.

It is now suggested that in both of these two last cases we have
processes in which there are not two, but three final fragments:

iH' + sW + 7^0 = 22He^ + o«' + T,,

(5)
(6)

22 Dee has just announced (Nature, 132, 818-819; Nov. 25, 1933) that these short-
range fragments are frequently paired. In doing the experiment he admitted the
primary protons into the expansion-chamber through a thin mica window, the
target being within.

CONTEMPORARY ADVANCES IN PHYSICS 137

the symbol o«^ in equation (5) standing for a neutron. When there
are three fragments, conservation of momentum no longer demands
that the available energy be equally divided among the three, but
admits of an infinity of distributions. It is not difficult to find the
highest fraction of Ti which either of the two alpha-particles in case
(5), or any of the three in case (6), may receive; this amounts to very
nearly one-half in the former, to two-thirds in the latter case.

In equation (5) the rest-masses of all the charged nuclei are known;
that of the neutron is still subject to some controversy, but if we

Fig. 15 — Various tracks produced during bombardment of lithium by deutons.
(Dee & Walton; Proceedings)

tentatively put Chadwick's value 1.0065 for it we get for (Ti — To)
the value 16 millions of electron-volts. 2"o again is negligible, so that
we are to compare half of this figure with the energy corresponding
to the range 7.8 cm. — the right-hand end of the sloping part of the
curve of Fig. 9 — which is 8.3 millions. The agreement is entirely
satisfactory. With boron the result is not so pleasing, for Ti by
equation (6) should be more than eleven millions, and two-thirds of
this differs rather seriously from the energy-value corresponding to the
end of the curve of Fig. 11, which is 6 millions. Kirchner got a
photograph in which three coplanar tracks of the same appearance

138 BELL SYSTEM TECHNICAL JOURNAL

diverge at mutual angles of 120° from a point in a boron target bom-
barded by protons, and Dee and Walton have noticed a number of
trios of paths springing from such a target, but without being quite
sure that they are not mere coincidences.^^

Having now met with a case in which there may not be a balance
between the two sides of such an equation as (6), we should now
pause to inquire what can be done about such cases. Of course,
such a disagreement might mean that the actual process is something
entirely different from the one postulated in the equation, but it may
not be necessary to make such a complete surrender of the theory.
In equations (1) to (6), it is everywhere assumed that all the energy is
retained by the material particles, in the form of kinetic energy or of
rest-mass. Suppose that the process described by one of these
equations, (6) for instance, is confirmed in every respect excepting
that the final kinetic energy of the fragments is found to be less, by
some amount Q, than the value of Ti computed from the equation.
One might then assume that the missing energy Q is radiated away
in the form of one or more photons. Alternatively one might assume
that the missing energy is retained by one of the material fragments
in the form of "energy of excitation"; the rest-mass of the fragment,
so long as it retained this energy and remained in the excited state,
would then be correspondingly greater than its normal rest-mass, and
the equation would be balanced if this abnormal value of mass were
inserted into it in place of the normal one. Such explanations are
frequently offered nowadays. They suffer, of course, from the
disadvantage of being too easy ; one can always postulate the necessary
photons or excited states to explain any observed positive value of Q.
But if they can ever be supported by independent proof of these
excited states or photons, they will become much more convincing.

Lithium and boron are by far the best-studied of nuclei, in respect
to their interactions with protons and deutons. It is true that our
knowledge of the distribution-in-range curves of the fragments is still
confined to comparatively low values of the energy of the bombarding
particles, values less than 300,000 electron-volts. With higher energies
it is to be presumed that the steps at the right-hand ends of the
curves in Figs. 8 and 9 would move to the right, to the extent pre-

^^ If in the case of boron bombarded by protons it be assumed that two of the He
nuclei fly off in directions making symmetrical angles (tt — 0) and (tt + 6) with the
direction of the third, the distribution-in-0 of the disintegrations can be deduced from
the curve of Fig. 14; it turns out that the most probable cases are those in which
6 = 60° nearly, and ail the three particles have nearly the same energy. A like
deduction may be made for lithium bombarded by deutons, the neutron playing the
part of third alpha-particle in the foregoing case; it is inferred that again the most
probable types of disintegration are those in which all three share almost equally in the
energy.

CONTEMPORARY ADVANCES IN PHYSICS 139

scribed by the increase of Tq in equations (1) and (2); and so should
the right-hand end of the sloping part of the curve in Fig. 9, and the
extremity of the curve of Fig. 11. There is an indication of the first
of these expected changes in the observation already quoted from
Lewis Livingston and Lawrence, of 14.8-cm. fragments ejected from
lithium by 1.33-MEV protons (page 133). We must wait for future
data to test the others, and to see what happens to the heights of the
steps and the general shape of the uninterpreted parts of the curves.
Already however we have data bearing on the so-called "disintegration-
function," or the relation of the total number of emitted fragments
to the energy of the bombarding particles.

To speak of "total number of fragments" is to suggest too much.
The present knowledge suffers from two limitations: the counts of
fragments are made with apparatus which does not enclose the target
completely and must be separated from the target by a screen, so that
the fragments counted are only those which start off within a limited
solid angle of deflections and have sufficient range to penetrate the
screen. One generally makes a tentative correction for the former
limitation, by assuming that the fragments go off equally in all
directions and multiplying the number observed by the factor 4x/co,
where w stands for the solid angle subtended by the detector as seen
from the target. This factor may well be wrong, but perhaps does
not vary seriously with the energy of the bombarding particles, so
that at least the trend of the curve may not be distorted. For the
latter limitation we have not the knowledge to make any allowance;
it must always be stated that the count is of fragments having more
than such-and-such a range, or such-and-such an energy. Every kind
of device for observing transmutation suffers from some such lower
limit, set either by the sensitivity of the device itself, or by the stopping-
power of the wall which bounds it.

With their dense streams of protons and exceedingly thin films
(page 113) Oliphant and Rutherford obtained the curves of Fig. 16:
the disintegration-functions of lithium and boron, with respect to
incident protons, up to proton-energies of some 200,000 electron-volts.
The wall between the target and the gas of the ionization-chamber had
an air-equivalent of 2.50 cm., and consequently the curves pertain only
to fragments having ranges greater than this.^^ The rise from the axis
is gradual, not abrupt; one might say that the shape of the curves
suggests that the protons have, not a definite capahility for transmuting
which begins suddenly at a critical energy, but a probability of trans-

-^ I hear from Dr. Oliphant that the trend of the curve for the short-range frag-
ments is just the same.

140

BELL SYSTEM TECHNICAL JOURNAL

muting which increases smoothly from zero (though this suggestion
might not occur to anyone not having foreknowledge of the current
theory!). The least energy at which transmutation is observable
should then depend entirely on the strength of the proton-stream and
the sensitiveness of the apparatus; von Traubenberg, with a stream

1300
1200
1100
1000
900

/
\

1
1
1

/
1

1
1

1 J
1 /

1 /

LL ,

800
700

1 /
1/

r LOG

1/
1/

600

400

1 1

1
/ 1

/ 1

300

Ll/

r 1
1
1

200

/ 1
/ /

BORON

>0 80 100 120 140 160

ENERGY IN EUECTRON-KILOVOLTS

180 200 220

Fig. 16 — Disintegration-functions of thin films of lithium and boron. (Oliphant

& Rutherford)

perhaps as strong as that of Oliphant and Rutherford, observed one
to three fragments per minute at 13,000 volts.

The curve of Fig. 17 extends very much further — all the way to
1.125 MEV — but was obtained with so thick a target of lithium
(lithium fluoride, to be precise) that the protons came to a stop in the

CONTEMPORARY ADVANCES IN PHYSICS

141

mass, and the disintegrations observed at any voltage might have been
produced by particles of any energy up to the maximum correspond-
ing to the voltage. It comes from the Berkeley school, the data being
procured chiefly by Henderson.^^ It refers only to fragments of ranges
superior to 5.32 cm., a grave limitation, accepted in order to make sure
that none of the primary protons could get into the detector (a Geiger
counter). From 400,000 volts onward, the curve of Fig. 17 conforms

OL 500

X COCKROFT AND WALTON

o LAWRENCE, LIVINGSTON

AND WHITE
• HENDERSON

.^ ,,^>

L^^"

.,,.^<<^

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

ENERGY IN ELECTRON-KILO VOLTS

Fig. 17 — Disintegration-function of lithium measured with a thick layer of lithium

fluoride. (Henderson)

to a simple and somewhat surprising assumption : viz. the assumption

that a proton of energy superior to 400,000 is neither more nor less

efficient in disintegrating lithium than a proton of only 400,000

electron-volts, and that the whole of the rise in the curve from this

voltage onwards is entirely due to the fact that the faster the proton,

the farther it dives into the target and the more chances it has to

^ The curve also fits the data of Cockcroft and Walton within the uncertainty of
experiment, due regard being had to the difference in the values of the solid angle
(letter from Dr. Henderson). In their work the screen between target and detector
had an air-equivalent of 3 cm. (letter from Dr. Cockroft). The curve of Fig. 8
shows that this had the same effect as Henderson's 5.32 cm.

142

BFXL SYSTEM TECHNICAL JOURNAL

impinge on a nncleus before it is slowed down and its energy reduced
beneath this particular value. The curve of Fig. 15 for lithium
should then become horizontal at abscissa 400. At lower voltages,
both curves concur in implying that the probability of disintegration
depends on the energy of the proton. I will revert to this topic in a
later article.

(

LlI
Q.

H
O
UJ

z
z

;

Ny'

UJ
O

Al TAF

?GET (E

iACKGRO

UND)

500 550 600 650 700 750

ENERGY IN ELECTRON-MLOVOLTS

Fig. 18 — Intensity of the mixture of neutrons and gamma-rays resulting when lithium
is bombarded by deutons. (Crane & Lauritsen)

There are also modes of disintegration of lithium by deutons and

by protons, in which neutrons and gamma-rays are emitted. These

have been observed in Pasadena by Crane, Lauritsen and Soltan.

Deutons are the more efficient of the two, but protons are sufficiently

potent to have enabled Crane and Lauritsen to trace the curves of

Fig. 18, in which the significant quantity is the difference between the

ordinates of the two.^" The ionization-chamber was walled inwardly

'"' Dr. Lauritsen writes me that the readings from which the lower curve is drawn
were unchanged when the high voltage was removed; presumably therefore they
represent the "background" due to the natural leaks of the electroscope. 1 am
indebted to his letter for other as-yet-unpublishe,d statements.

CONTEMPORARY ADVANCES IN PHYSICS 143

with paraffin, to accentuate the effect of the neutrons; it was however
found that the readings were not considerably lessened when the
paraffin coating was absent, and consequently Lauritsen infers that
most of the effect is due to gamma-rays proceeding from the bombarded
atoms. This inference is sustained by the fact that when the rays
responsible for the effect are caused to pass through leaden screens,
the ionization falls off exponentially with the thickness of the lead;
and the value of the exponent suggests that the energy of the photons
is about 1.5 MEV. One can easily think of a process whereby deutons
might evoke neutrons from lithium nuclei :

iH2 + 3Li- + To = 22He^ + on' + Tu (7)

but with protons no plausible interaction comes readily to mind.
Perhaps there is a two-stage process, the protons producing the
reaction described by equation (1), the resultant He^ nuclei striking
other lithium nuclei and evoking neutrons. Or perhaps the neutrons
and the gamma-rays alike result from the same processes as produce
the groups of short-range alpha-particles revealed in Fig. 8. Questions
of this intricate kind will probably predominate in the study of trans-
mutation, in the years to come; and experiments on thin films will
play a very important part in settling them, both because the likelihood
of two-stage processes will be reduced, and because it may be possible
to learn which isotopes are involved.

Little indeed is definitely known about the disintegration, by
protons or deutons, of any other elements than lithium or boron.
Charged fragments have been observed proceeding, in relatively small
but yet appreciable number, from bombarded targets made of a great
variety. But in many of these cases they may be due, so far as any
of the observations tell, to a minute contamination of the target by
boron derived from the glass of the enclosing tube; and the danger
of this possible source of error was vividly brought out by Oliphant
and Rutherford, when at first they observed such fragments, but
ceased altogether to observe them when the original glass of their
tube was replaced by a special boron-free variety! Beryllium and
fluorine are the only elements, other than lithium and boron, of which
these experimenters were sure of detecting fragments; for those of
fluorine they were able to plot a disintegration-function and a distribu-
tion-in-range, which differed sufficiently in aspect from those of
lithium and boron to exclude the possibility that these might be
responsible; those of beryllium were too scanty for such tests. The
elements with which they got no charged fragments, or only a few
per minute, were the following: Fe, O, Na, Al, N, Au, Pb, Bi, Tl, U,

144 BELL SYSTEM TECHNICAL JOURNAL

Th. But their observations were confined to protons of relatively
low energy-values, — their upper limit was little over 200,000 electron-
volts — and do not prove that faster particles are incapable of trans-
mutation. The Berkeley school has already published a number of
observations made with protons of energies ranging up to 710,000,
and with deutons of energies attaining the unprecedented height of
3 MEV; and they find fragments in abundance from a wide diversity
of targets.

Beryllium deserves a special paragraph, since it yields neutrons
when bombarded, whether with alpha-particles from radioactive
bodies; or with helium ions extracted from a discharge and endowed
artificially with energies of 600,000 electron-volts and upward; or with
deutons. The first of these processes is the one which led to the
discovery of the neutron; the second, which incidentally marks the
first employment of artificial alpha-particles (since these helium ions
are alpha-particles in all but origin, except for the unimportant
difference that each possesses an extra-nuclear electron while it is
approaching the target) is a recent achievement of the Pasadena
school (Crane, Lauritsen and Soltan) ; the third was achieved both at
Pasadena and at Berkeley. These three processes are now in rivalry
with one another, and it remains to be seen which will be producing
the greatest number of neutrons, a year or five years hence. It is
still very doubtful how the third takes place: perhaps the deuton
merges with the beryllium nucleus, as in the other cases the alpha-
particle is supposed to do (page 155), or perhaps it knocks a pre-
existent neutron out of the beryllium structure and goes unaltered on
its way. This too is a problem for the future, and one in the solving
of which the charged fragments likewise observed will probably play
a part.

The deuton itself is in all probability a complex particle; might it
not be shattered in impinging against a nucleus, especially some heavy
nucleus? This is the interpretation offered by Lawrence of the fact
that in sending streams of deutons against targets of several different
kinds, he observed charged fragments which were protons (not alpha-
particles !) forming a group having a definite range and a definite
energy not depending at all on the substance of the target. With
1.2-MEV deutons this characteristic energy of the protons is 3.6 MEV.
A singular rule governs this quantity: if the energy of the bombarding
particles is increased, that of the protons goes up by just the same
amount — deutons of energy (1.2 -f x) MEV evoke protons of energy
(3.6 + x) MEV. The rule has been verified for values of x up to 1.8,
Such a rule is just what one would expect, were there no other frag-

CONTEMPORARY ADVANCES IN PHYSICS 145

merits than the protons, excepting fragments of such great mass that
they could take up the necessary momentum without taking an
appreciable amount of kinetic energy. The heavy nucleus by itself
is able to do this. However there are also neutrons, of which the
energy is sufficient to let them be detected, and therefore by no means
negligible. This is gratifying for the theory, inasmuch as if a proton
is separated from a deuton, the residue should be a neutron (or else
another proton and a free electron) ; but one is then obliged to assume
that the neutron always takes the same kinetic energy, whatever
that of the impinging deuton may have been. This seems rather odd,
but nothing prohibits it. Streams of alpha-particles have been sent
against compounds ("heavy water') containing deuterium in abun-
dance, but as yet no neutrons have been detected coming off.

Transmutation by Impacts of Alpha-Particles ^^

Impact of an alpha-particle against a nucleus may result in the
springing-off of one or more (or none) of four kinds of corpuscles:
protons, photons, neutrons, positive electrons.

Trans77iutation ivith production of protons

This is the earliest-discovered type, of which I told at length in
"Transmutation." The discovery was made by Rutherford in 1919
in experiments on nitrogen. At present the Cavendish school considers
that this mode of transmutation has been proved for thirteen elements,
none of atomic number greater than 19: the list comprises B, N, F,
Ne, Na, Mg, Al, Si, P, S, CI, A, K. The most frequently and fully
studied cases are those of boron, nitrogen and aluminium.

The evidence that the fragments are protons is rather variegated.
In some cases this has been proved by deflection-experiments;^^
recently it has been proved in some other cases by measuring both
the range of the fragments and the ionization which they individually
produce in a shallow chamber or a deep one (page 125) ; some observers
are able to tell the scintillations due to protons from those which are
due to alpha-particles.

Integral distribution-in-range curves of the fragments have been
obtained for boron, nitrogen, fluorine, sodium, magnesium, aluminium
and phosphorus. Most of them show more or less conspicuous
plateaux, of which the most magnificent appear in the celebrated
curves of Pose for aluminium, reproduced in "Transmutation"

^^ An expanded version of this section, with citations of additional data and
reproductions of some curves, appears in the Physics Forum of the Review of Scientific
Instruments for February 1934.

^^ "Transmutation," pp. 636-640, B. S. T. J., Oct. 1931.

146 BELL SYSTEM TECHNICAL JOURNAL

(Figs. 6, 7) ; from this there are all gradations of distinctness downward,
ending with cases in which it is uncertain whether the ideal curve
would be a smoothly-descending one, or would have a succession of
short plateaux which in the actual curve are rounded off into indis-
tinguishability.

By "ideal curve" in the foregoing sentence I mean, as heretofore
(page 130), that which would be obtained with an infinitely narrow
beam of fragments proceeding in a single direction and produced by
alpha-particles all of a single speed and proceeding in a single direction.
I must also add that many thousands of fragments should be counted,
as otherwise the results are likely to be distorted by statistical fluctu-
ations. It appears that in most of the experiments with bombarding
alpha-particles, the departure from the ideal is much more considerable
than in the best of the experiments with bombarding protons. The
targets are usually so thick that the speeds of the alpha-particles
vary considerably as they go through, and often so thick that these
are swallowed up and every energy of bombarding particle, from the
initial maximum down to zero, is represented among the impacts.
This matters much more than it does with protons, because here the
energy of the primary particles is often much greater than that of the
fragments, and a small percentage variation of the former may entail
a big one of the latter. The solid angles subtended by the exposed
part of the target as seen from the source of the alpha-rays on the
one hand, from the detector on the other, are frequently both large.
This is particularly serious, because it appears that the ideal distribu-
tion-in-range curve would vary with the angle between the directions
of the impinging particle and of the fragment. In some experiments
the number of fragments observed has been too small to be immune to
statistical fluctuations, and it is surprising that the plateaux in Pose's
curves should be so clear despite this handicap.

Where two or more observers have studied a single element, there is
generally enough concordance among their statements to assure the
onlooker that at least the major groups of protons are recognizable.
The prettiest case thus far is that of nitrogen: three researches on the
integral distribution-in-range curve agree in showing a sharply-marked
group of range about 17.5 cm (for protons ejected forward by full-speed
alpha-particles from polonium, energy 5.3 MEV). The flattest
plateau and sharpest step are to be seen in a curve by Chadwick Con-
stable & Pollard, who approached very nearly to the ideal experiment in
one respect, by using a stratum of nitrogen so thin that its air-equiva-
lent was only 3 mm. All the protons of range superior to about 6 cm.
belong to this group; there is another of inferior range, lately discovered

CONTEMPOR.ARY ADVANCES IN PHYSICS 147

by Pollard. Phosphorus and sodium have been studied only by Chad-
wick Constable &. Pollard, who find for the former a single group, for
the latter a smoothly-descending int«tral curve which may betoken
total absence of groups, or may be resolved, by some future and closer
approach to the ideal curve, into a close succession of bends and corners.
The four remaining elements — B, F, Mg, Al — show at least three groups
apiece, and indeed Chadwick and Constable deduce four pairs of
groups for aluminium and three for fluorine. To illustrate the degree
of concurrence between different observers, I quote the values for the
groups of aluminium — that is to say, values of the ranges of the protons
belonging to these groups, ejected forward by 5.3-MEV alpha-particles
— from the four authorities. Pose gives 28.5, 49.6, and 61.2 (cm of air-
equivalent) ; Steudel, 33, 49, 63; M. de Broglie and Leprince-Ringuet,
30, 50, 60; Chadwick and Constable give 22, 26.5, 30.5, 34, 49, 55, 61,
66. More detailed comparisons had best be left to those who have
practice in this field.

While nearly all of the data have been obtained by other methods
than that of the expansion-chamber, a few beautiful pictures have
been taken in which there appears the track of an alpha-particle
passing through nitrogen, and this track is seen to end at a fork.^^
One of the tines of the fork is a long thin track, apparently that of a
proton; there is only one other, and this is short and thick. It is
inferred that these reveal the only fragments which there are, and that,
in the usual though somewhat objectionable phrase, the alpha-particle
has fused with the residual nucleus. The process is then expressed
by the equation :

tN^^ + 2He^ + To = sQi^ + iRi + T„ (8)

the symbols being chosen according to the same principles as in
equation (1). It is commonly assumed, though in no other case with
such good evidence, that this happens in most if not in all cases, so
that when a nucleus of atomic number Z and mass-number A is
transmuted by an alpha-particle, the process often is :

zM-" + 2He^ + To = z+iAH+3 + iH^ + Ti, (9)

with an obvious symbolism. This is called "disintegration with
capture" (though it is the case in which the objection to the name
"disintegration," page 117, is gravest). The other conceivable case of
"disintegration without capture" would be described thus:

zM-^ -f 2He^ + To = z-iM-^-i + iHi -f aHe^ + T,. (10)

^'' "Transmutation," Figs. 10 and 11.

148 BELL SYSTEM TECHNICAL JOURNAL

Disintegration-with-capture is very advantageous for the theorist,
since when there are only two fragments after the interaction the
principle of conservation of momentum suffices to determine the
kinetic energy of either in terms of that of the other and that of the
alpha-particle. In equation (9), Tq stands for the kinetic energy of
the alpha-particle, Ti for the sum of the kinetic energies of the proton
and the residual fragment, which call Tp and Tr respectively. Now
excepting in the cloud-chamber experiments, it is only the proton
which is detected, and therefore only Tp can be estimated from the
data; but if the disintegration is by capture, then Tr and consequently
Ti can be deduced from Tq and Tp. If however there are three or
more final fragments, measurement of Tp is not sufficient to determine
T\. Also even in the case of disintegration-by-capture there will be
uncertainty if the transmuted element is a mixture of two or more iso-
topes, since the value of Tr corresponding to an observed Tp will de-
pend on the mass of the atom which is transmuted.

In a case of disintegration-by-capture, the simplest possible assump-
tion is that {T\ — Tq) has a perfectly definite value, independent of Tq\
there is conversion of a definite amount of kinetic energy into rest-mass
(or vice versa), whatever the velocity of the alpha-particle may be.
This may be tested by varying To; it may also be tested to some extent
by observing protons ejected in various directions (relatively to the
initial direction of the alpha-particles) since although the sum of Tp
and Tr (which is T\) should be the same for all of these protons those
two quantities individually should vary, and Tp in particular should
depend in a definite manner on the direction of the protons. Yet in
nearly all such tests, the target is so thick that the alpha-particles im-
pinging on various nuclei have very various speeds. How then shall
we know which speed of proton to associate with which speed of alpha-
particle, which value of Tp belongs with which of TqI One naturally
begins by assuming that the fastest of the primary particles produce
the fastest of the protons. But plausible as this assumption seems at
first, there are several cases known in which it is not true: cases in
which a definite group of protons is evoked by alpha-particles of a
definite interval of speeds, and neither faster nor slower particles are
capable of producing them.

This phenomenon of " resonance, " as it is called,^" was first observed
by Pose in the experiments on aluminium to which many pages were
devoted in "Transmutation." It is evidently an important quality
of nuclei, destined to be prominent in experiment and theory both.

5° There is a tendency to use the term "resonance" to express the mere existence
of groups, irrespective of whether they are evoked by alpha-particles of narrowly
limited speeds. This is to be deprecated.

CONTEMPORARY ADVANCES IN PHYSICS 149

This makes it desirable to consider at some length how resonance may
be detected. There are the following ways:

(a) When the target is thick, one may vary the energy Kq which
the particles possess when they strike the target-face Kq (usually by
varying the density of gas between the target-face and the source
of the alpha-particles) and plot the integral distribution-in-range curve
for many different values of Kq. Let us suppose that there is a certain
proton-group evoked only by alpha-particles having energy between
Ka and Kh, the notation being so chosen that Kh < Ka < Kq. Then it
will be found that as Ko is lowered, the step and plateau which reveal
the group will remain unaltered until Ko drops below a certain critical
value (to be identified with Ka) after which they will fade out.

(b) In the foregoing conditions, one may use a very thin ionization-
chamber and plot instead of the integral distribution-in-range curve a
curve of the sort in Fig. 10, or the sort described on page 129 of which
the ordinate stands for the number of fragments producing more than
a certain chosen amount of ionization in the chamber. There will be
various peaks in the curve corresponding to various groups, and if any
of these is produced by "resonance" it will at first remain unaltered
and then gradually disappear as Ko is lowered.

(c) When targets thin enough to be completely traversed by the
alpha-particles are available, one may leave Kq unchanged and increase
the thickness t of the target. The energies of the impinging particles
in a target then vary from i^o down to a minimum value Ki which
depends on /. If curves of any of the foregoing kinds be plotted for
various values of Ki, and if any of the groups is produced by resonance,
then the step or the peak corresponding to this group may be absent
when Ki is high (i.e. with the thinnest target) and will then make its
appearance when Ki is lowered past a certain critical value (again to
be identified with Ka).

(d) If the target is so very thin that the loss of speed suffered by
the alpha-particles in going through is negligible, and Ki is sensibly
equal to i^o. then when Kq is varied the groups should appear and
disappear when it becomes equal to Ka and Kb, respectively.

(e) Without subjecting the fragments to any analysis, one may
simply measure the total number thereof (or rather, the total number
having ranges superior to some fixed minimum) as function of i^o-
Suppose the target to be thick; then, if all the proton-groups are
evoked by resonance, the curve should display a sequence of steps
and plateaux; if in addition to such there are groups which are evoked
by particles of any energy over a wide interval, the steps need not
vanish, but the plateaux should slope upward and may be curved.

150 BELL SYSTEM TECHNICAL JOURNAL

If the target is very thin (in the sense of the previous paragraph)
the curve ought to show a peak for each group. Such curves, by the
usage of page 139, may be styled "disintegration functions" (the term
"excitation-function" is also used).

(/) Finally, when the target is thick the mere existence of sharp steps
in the integral distribution-in-range curves, may be taken as a sug-
gestion of resonahce, since if a group were evoked by alpha-particles
of a wide range of energies it would probably have a broad distribution
of speeds. But this is not a very strong argument by itself.

Despite this great variety of ways of testing for resonance, the situa-
tion is still confusing and confused.

Aluminium has been the object of most of the tests, doubtless be-
cause it figured in Pose's discovery. He used methods (a) and (c)
and found resonance distinctly and even vividly displayed by tlte
60-cm. and the 50-cm. group, and not at all by the 25-cm. group. Chad-
wick and Constable used {a) and {b), and concluded that there is
resonance for six at least of their eight groups, the two members of a
pair appearing and disappearing together. (The remaining pair was
elicited by alpha-particles of a limited interval of energy- values extend-
ing from a lower limit Kh to the highest value of Kq which they had
available.) They also used (e) with a very thin sheet of aluminium
(air-equivalent 0.8 mm.) and got a curve with two well-defined peaks.
But Steudel also had recourse to method (e), and the curve he got
swept smoothly upward ; it is true that his target was notably thicker
(air-equivalent 5.2 mm.) and yet one would not expect such a thickness
to blot out the peaks if they exist. Harder yet to explain away is the
evidence of M. de Broglie and Leprince-Ringuet, who made test {d)
with sheets of aluminium of air-equivalent 2.5 mm., and observed all
three of Pose's groups over a wide range of values of Kq. — As for the
other elements: boron and fluorine and magnesium have all been
tested by method (a) , and there are strong indications of resonance for
all three, strongest for fluorine. Nitrogen has been studied by Pollard
with a modification of (e), and he finds that resonance is displayed by
the 6-cm. group but not by the stronger and better-known group of
longer range.

Evidently this is a field which yearns for further cultivation, with
more powerful sources of transmuting particles to make possible the
use of narrower and more homogeneous beams of these, narrower pen-
cils of fragments and thinner strata of matter. The discovery of the
capacity of protons to transmute has probably diverted from it some of
the attention which otherwise it would by now have received, but the
lost ground will doubtless be made up in the course of years, after the

CONTEMPORARY ADVANCES IN PHYSICS 151

developments which that discovery has hastened shall have brought
about the generation of streams of artificial alpha-particles more
numerous by far than the natural ones. Meanwhile we must be con-
tent with scanty data and with fragmentary tests of the important
question already mentioned: whether the energy transformed from
rest-mass to vis viva or reversely — the quantity here denoted by
{Ti — To), elsewhere commonly by Q, designated in German as the
Tbnung of the process — is a definite and characteristic quantity.

Certainly about resonance is essential to these tests ; for if resonance
exists, we have to correlate the energy of a group of protons with that
particular energy of the alpha-particles which evokes the group; but if
resonance does not occur, then probably the best we can do is to cor-
relate the energy of the fastest of the ejected protons of a group with
that of the fastest of the impinging particles — and if we make the latter
guess w^hen it ought not to be made, there will be trouble! Perhaps
the most impressive evidence is that available for aluminium. Chad-
wick and Constable evaluated {Ti — To) for all of their eight groups:
the si.x for which they demonstrated resonance, and the two which were
evoked by alpha-particles of a limited interval of energies extending up
to the highest which they used, which was 5.3 MEV. They find that
(Ti — To) has a common value of +2.3 MEV for four of their groups —
to wit, the longer-range members of their four pairs — and a common
value of zero for the other four. Haxel plotted the integral distribu-
tion-in-range curves for the protons ejected by alpha-particles of sev-
eral yet higher energies, running up almost to 9 MEV; he detected two
groups; they did not display resonance, but he correlated the highest
energy represented in each wdth the highest represented among the
impinging particles, and he too found +2.3 MEV and zero for {Ti — To)
in the two cases ! ^^ Blackett analyzed eight examples of transmutation
of nitrogen observed with the cloud-chamber (here he had the unique
advantage of being able to observe the track of the residual nucleus
and estimate its energy) and he reported for {Ti — To) a mean value of
— 1.27 MEV with a mean deviation of 0.42 from the mean. Future
confirmation awaited this work also: Pollard, analyzing his integral
distribution-in-range curves, made a computation of (Ti — To) for the
6-cm. group which exhibits resonance, and another for the 17.5-cm.
group which does not, correlating in this latter case the energy of the
fastest protons with that of the fastest alpha-particles; the results were
-1.32 and -1.26 MEV.

'^ The precision of these values can hardly be estimated from what Chadwick and
Constable say, but some idea of it can be gained from a graph in Haxel's article,
ZS.f. Phys. 83, p. 335 (1933), and loc. cit. footnote 27.

152 BELL SYSTEM TECHNICAL JOURNAL

Such are the cases where there is the strongest proof for the twin
doctrines that disintegration is by capture, and that a definite amount
of energy is transformed between rest-mass and vis viva. The reader
will have noticed in the latter case, that {Ti — To) appeared to be the
same for a group which exhibits resonance and for another group which
does not. This if certain may be taken to mean, that a particular
group of protons — one may speak more graphically, and say: a par-
ticular proton in a particular level of the nitrogen nucleus — can be ex-
tracted by alpha-particles of a narrowly-limited range of energies be-
tween critical energy-values Ka and Kb, and can also be extracted by
alpha-particles of any energy superior to a third critical value Kc which
is greater than Ka and Kb. There is a good interpretation of this
notion in the contemporary theory, which I reserve for the next article.
It will also have been noticed that two different values of (Ti — To)
were given for a single case, that of aluminium (there are also two for
fluorine). This is to be taken as meaning that the residual nucleus
may be left in either of two conditions, one of which may be the normal
state, while the other must be an excited state (page 138). One then
infers that the nucleus when left in the excited state will presently go
over to the normal state, emitting a photon having an amount of energy
equal to the difference between the two values of {Ti — To). It is
very tempting to suppose that the gamma-rays known to be emitted
from some elements during alpha-particle bombardments have this
origin, but the measurements are not yet precise enough to prove this.*^

In a case of disintegration-by-capture, the residual nucleus denoted
by z+iM^"*"* in equation (9) might or might not be exactly the same as
the nucleus of the known chemical atom (if such there be) of atomic
number (Z + 1) and mass-number (A +3). Can this be tested by
comparing the rest-mass of the former with the mass of the latter as
measured by Aston or Bainbridge? Unfortunately nothing of value
can be concluded unless the atoms z+iM"^"^^ and ^M^ have both had
their masses determined with an accuracy permitting them to appear in
the Table on page 109 ; and on inspecting this table one finds (with some
surprise) that this is true for only one of the known processes, viz. the
transmutation of fluorine. Assuming disintegration to be with cap-
ture, the process would be the following:

9F» + 2He^ = loNe^^ + ^h^ -f- {T, - To) (11)

Putting for {Ti— To) the value -M.67 MEV given by Chadwick and
Constable, and for the rest-masses of the nuclei the values given in the

''^ Heidenreich has analyzed the data for boron, and concludes that they permit
of this interpretation. {ZS.f. Phys. 87, 675-693; 1933.)

CONTEMPORARY ADVANCES IN PHYSICS 153

Table, we get 23.002 for the left-hand member and 23.0043 for the
right-hand member. The agreement is within the uncertainty of the
data; so also would it have been, had {Ti — To) been ignored. Its im-
portance is perhaps enhanced by the fact that it is ex post facto: the
mass of Ne^^ was inaccurately known at the time of the experiments of
Chadwick and Constable, and there was ostensibly a disagreement.

I repeat that it is not proved that transmutation occurs in every case
by capture; and an isolated value of {Ti — To), such as one often sees
computed from a single observation on a particular group evoked by a
particular beam of alpha-particles, is not necessarily valid.

Transmutation with production of neutrons
This mode of transmutation has been proved, according to the
Cavendish school and the Joliots, for the elements Li, Be, B, F, Ne, Na,
Mg, and Al. The outstanding cases are those of beryllium and boron,
with lithium and fluorine following after. Negative results have been
reported by the Joliots for H, C, O, N, P and Ca, and there is no record
of a positive result for He. Positive results have been reported for
quite a number of elements both light and heavy by the Vienna school.
There is nothing which can properly be called a distribution-in-range
curve for neutrons; but there is something which is potentially as use-
ful— the integral distribution-in-range curve of the protons emanating
from a thin layer of matter rich In hydrogen, placed between the source
of the neutrons and the detector. If one can measure the speed of a
proton recoiling in a known direction from the impact of a neutron,
one can deduce the speed of the neutron ; in particular, if one can meas-
ure the speeds of the protons projected straight forward by central im-
pacts of the oncoming neutrons, one may consider their speeds as
practically the same as those of the neutrons themselves. ^^ It is thus a
proper procedure to obtain the integral distribution-in-range curve of
the protons projected forward, and convert it into a distribution-in-
energy curve which is that of the protons and the neutrons alike. It
has however not been an easy procedure, on account of the sparseness
of the available sources of neutrons and hence of the streams of recoil-
ing protons. Chadwick has published a solitary curve of this sort,
relating to the neutrons from beryllium ejected by the alpha-particles
of polonium; and Dunning has obtained a curve displaying good
plateaux and steps, relating to the neutrons from beryllium ejected by
yet faster alpha-particles.^* Steps and plateaux, as heretofore, signify
groups of protons and consequently groups of neutrons. Feather has

33 Cf. Part I, page 300.

'■' To be published in the article mentioned in P"ootnote 27, and by Dr. Dunning
himself.

154 BELL SYSTEM TECHNICAL JOURNAL

achieved the feat of taking and examining no fewer than 6900 cloud-
chamber photographs in order to deduce the distribution-in-speed of
neutron-streams from the tracks of the recoiUng nuclei of various kinds
of atoms. Most observers publish no curves, but give only verbal ac-
counts in which they state the thickness (in air-equivalent) of the
intercepting screens athwart the proton-beam, for which they observed
a notable falling-off of the strength of that beam; or else they state
what groups they believe in, inferring them presumably from observa-
tions of that type. This makes tiresome and unsatisfactory reading.

Much of recent research is meant to detect the very fastest neutrons
emitted from a given element, for a reason which will presently be
obvious if it is not already. Chadwick gives 3.35 MEV for the energy
of the fastest neutrons ejected from boron by polonium alpha-particles,
and 12 MEV for those similarly ejected by beryllium, while Dunning
gives 14.3 MEV for those which beryllium emits when bombarded by
the somewhat faster alpha-particles from radon.

Curves called "disintegration-functions," or more commonly "exci-
tation-functions," have been plotted several times for the neutrons
from beryllium and once at least for those from boron. One must
realize an important distinction between them and the curves obtained
when the fragments are alpha-particles or protons, as in Figs. 16 and
17. When the fragments are charged particles, it is practically certain
that all of them which reach the detector at all are duly detected.
When the fragments are neutrons it is certain that the only ones de-
tected are those which strike protons (or other nuclei) hard enough and
squarely enough to give them a considerable amount of energy and
enable them to produce a good many ions in the ionization-chamber;
and it is equally certain that those constitute but a small fraction of the
total number of neutrons, most of which go through the expansion-
chamber unperceived. Would that this were at least a constant frac-
tion! we could then rely on the shape of the so-called excitation-curve,
while realizing that all its ordinates must be multiplied by some un-
known but constant factor. But we must not suppose even this; it is
practically certain that the factor varies with the speed of the neutrons,
and hence in all probability with the speed of the primary alpha-
particles; and hence the so-called excitation-curve must be distorted
from the true curve of number-of-atoms transmuted versus energy-of-
alpha-particles. (Also the distribution-in-range curves must be dis-
torted.)

With these severe limitations in mind, one may consider the pub-
lished excitation-functions. The most striking are those obtained with
very thin films of beryllium, one by Chadwick and one by Bernardini,

CONTEMPORARY ADVANCES IN PHYSICS 155

which agree in showing a rather sudden rise of the curve from the
horizontal axis, then a peak, then a valley and then a sweeping rise.
It is hardly likely that the peak and the valley are entirely due to dis-
tortion of a truly smoothly-rising curve by the aforesaid agency; and
the argument of paragraph (e) of page 149 leads us to infer a group of
neutrons displaying resonance, in addition to other neutrons for which
perhaps there is no resonance. Curves obtained with thick targets of
beryllium or of boron have conspicuous steps, carrying the same im-
plication. Those for boron (Chadwick and the Joliots) and some of
those for beryllium (Rasetti, Bernardini) suggest but a single group,
but there are other curves for beryllium suggesting two (in recent work
of Chadwick's) and even four (Kirsch and Slonek). Thus, although
the first four tests of resonance w^hich I listed above (page 149) have as
yet remained untried for emission of neutrons, the fifth has given some
pretty convincing evidence in its favor.

It is always assumed that transmutation with emission of a neutron
is a case of disintegration-by-capture, though no one has proof of this
yet. The imagined process may be symbolized thus:

zM-^ + 2He4 + T, = z+2N^^3 + ^„i j^x, (12)

Such equations as this are used for evaluating the rest-mass of the
neutron, it being assumed that the rest-mass of the residual nucleus
z+2^^^^ is identical with that of the nucleus of the atom of mass-number
{A -f 3) and atomic number (Z + 2). One encounters at once the
difficulty that there are neutrons of a wide range of speeds, and conse-
quently a wide range of values of Ti. It is necessary to assume that the
slower neutrons leave behind them a nucleus in an excited state
(page 138) and that only the very fastest leave behind them the normal
nucleus which is to be identified with that of the isotope {A -\- 3) of the
element (Z -f- 2). Doing this, Chadwick got consistent values for the
mass of the neutron from the observations on boron and on lithium,
assuming the nucleus M of equation (12) to be that of B^^ and that of
Li'^ respectively.^^ To obtain a consistent value from the neutrons of
beryllium, one would have to observe some at least having an energy as
great as 12 MEV (when To = 5.3 MEV). Those observed in the earlier
work on beryllium were all much too slow. One of the driving motives
of recent research has been the desire of finding at least a few^ of ade-
quate energy; and it appears that this desire has at last been fulfilled.

'* Were we to assume B'" and Li'^, the nucleus N would correspond to an isotope as
yet unknown; this is a powerful but not an absolutely imperative argument against
these choices. There is also the question of whether, if resonance occurs, the right
correlation is being made between values of T\ and values of T^ (page 151). — The
equation for the transmutation of boron has been worked out in Part I., pp. 323-324.

156 BELL SYSTEM TECHNICAL JOURNAL

To guess at the total number of neutrons emitted (say) from beryl-
lium it is necessary to know the excitation-curve and to make an esti-
mate of the factor aforesaid. I confine myself to quoting from Chad-
wick: "The greatest effect is given by beryllium, where the yield is
probably about 30 neutrons for every million alpha-particles of polonium
which fall on a thick layer,"

Transmutation with production of positive electrons

This mode of transmutation, as I mentioned earlier, has been ob-
served by the Joliots with Be, B and Al, the primary corpuscles being
polonium alpha-particles. Nothing has yet been published about
distribution-in-range or disintegration-function. Positive electrons
of energy as high as 3.1 MEV have been observed proceeding from
aluminium.

Aluminium thus affords a case of an atom which under alpha-particle
bombardment may emit from its nucleus a particle of any of three
kinds: a proton, a neutron, a positive electron. It has been suggested
by Joliot that there is actually only one process, in which a proton
emerges either intact, or else split into a neutron and a positive electron
which are its hypothetical components. If this can be verified it will
have important bearings on various fundamental questions, including
that of the mass of the neutron.^*' Boron also emits particles of all
three kinds, but here the situation is complicated by the possibility
that not all of the three proceed from the same isotope.

Transmutation by Neutrons

Transmutation by neutrons has been observed only with the Wilson
chamber, and therefore rarely: there are a few scores of recorded cases,
the fruit of twenty or thirty thousand separate photographs taken
some by Feather at the Cavendish, some by Harkins and his colleagues
at Chicago. What is observed is a pair of tracks diverging from a
point in the midst of the gas contained in the chamber; it is inferred
that the (invisible) path of a neutron extends from the neutron-source
to the point of the divergence, and that the observed tracks are those
of two fragments of a nucleus which that particle has struck. "Frag-
ment ' ' must be taken in the generalized sense of page 117: the substance
of the neutron may be comprised in either or both of the two. Each
case must be separately analyzed, taking into account the directions
and the ranges of the fragments (it is here that the question of the
range-z^5-energy relations of massive nuclei, footnote 20, becomes
crucial). It is possible to infer that in many cases the neutron is ab-

^^ See the reference in Footnote 27.

CONTEMPORARY ADVANCES IN PHYSICS 157

sorbed into the fragments — "disintegration with capture" — and even
to estimate {Tx — To), which turns out to be usually if not always
negative. There are some difficulties here, since in certain cases the
process which is observed seems to be the converse of one of the well-
known processes of generating neutrons, and yet (Ti — To) does not
appear to have values equal in magnitude and opposite in sign for the
two. The most startling feature of transmutation by neutrons is,
that it occurs with nuclei which seem to be immune to other transmut-
ing agents, notably carbon and oxygen. Other elements with which it
occurs are nitrogen, fluorine, neon, chlorine and argon.

Acknowledgments

I am greatly indebted to Monsieur F. Joliot, Professor E. O. Law-
rence, Dr. J. R. Dunning and Dr. P. I. Dee for providing me with
prints of several of the photographs which appear in this article (Figs.
1, 4, 5, 6, 14, 15); and to Dr. Dunning for criticism and advice in
respect to several sections of the text.

References

Transmutation by Protons and Deutons
Cavendish school:

J. Cockcroft & E. T. S. Walton: Proc. Rov. Soc. A129, 477-489 (1930); 136,

619-630 (1932); 137, 229-242 (1932).
P. I. Dee: Nature 132, 818-819 (25 Nov. 1933).
P. I. Dee & E. T. S. Walton: Proc. Roy. Soc. A141, 733-742 (1933).
M. L. E. Oliphant & E. Rutherford: Proc. Roy. Soc. A141, 259-281 (1933).

The same with R. B. Kinsey: ibid. 722-733.

Berkeley school:

M. C. Henderson: Phys. Rev. (2) 43, 98-102 (1933).

E. O. Lawrence & M. S. Livingston: Phys. Rev. (2) 40, 19-35 (1932).

Letters and abstracts by E. O. Lawrence, M. S. Livingston, M. G. White,
G. N. Lewis, M. C. Henderson: Phys. Rev. (2) 42, 150-151, 441-442 (1932);
43, 212, 304-305, 369 (1933); 44, 55-56, 56, 316-317, 317, 781-782, 782-
783 (1933).

Other schools:

H. R. Crane, C. C. Lauritsen & A. Soltan, Phys. Rev. (2) 44, 514 (1933) (effect

of He+ ions); ibid. 692-693; Crane & Lauritsen, ibid. 783-784; 45, 63-64

(1934).
C. Gerthsen: Naturwiss. 20, 743-744 (1932).

F. Kirchner: Phys. ZS. 33, 777 (1932); 34, 777-786 (1933); with H. Neuert,

34, 897-898 (1933). Sitzungsber. d. kgl. Bdyrischen Akad. 129-134 (1933).
Naturwiss. 21, 473-478, 676 (1933).
H. Rausch v. Traubenberg, R. Gebauer, A. Eckart: Naturwiss. 21, 26 (1933);
ibid. 694.

Transmutation by Alpha-Particles
Transmutation with emission of protons:

P. M. S. Blackett: Proc. Roy. Soc. A107, 349-360 (1925).

W. Bothe: ZS. f. Phys. 63, 381-395 (1930); Atti del convegno di fisica nucleare,

Roma, 1932.
W. Bothe & H. Franz: ZS.f. Phys. 43, 456-465 (1927); 49, 1-26 (1928).

158 BELL SYSTEM TECHNICAL JOURNAL

W. Bothe & H. Klarmann: Natnrwiss. 35, 639-640 (1933).

M. de Broglie & L. Leprince-Ringuet: C. R. 193, 132-133 (1931).

J. Chadwick, J. E. R. Constable & E. C. Pollard: Proc. Roy. Soc. A130, 463-489

(1931).
J. Chadwick & J. E. R. Constable: Proc. Roy. Soc. A135, 48-68 (1932).
K. Diebner & H. Pose: ZS.f. Phys. 75, 753-762 (1932).
W. D. Harkins: with R. W. Ryan, J. Am. Chem. Soc. 45, 2095-2107 (1923);

with H. A. Shadduck, Proc. Nat. Acad. Sci. 2, 707-714 (1926); with A. E.

Schuh, Phvs. Rev. (2) 35, 809-813 (1930).

0. Haxel: ZS.f. Phys. 83, 323-337 (1933).

F. Heidenreich: ZS.f. Phvs. 86, 675-693 (1933).
(;. Hoffmann: ZS.f. Phys. 73, 578-579 (1932).
C. Pawlowski: C. R. 191, 658-660 (1930).

E. C. Pollard: Proc. Roy. Soc. A141, 375-385 (1933).

H. Pose: Phys. ZS. 30,' 780-782 (1929); 31, 943-945 (1930). ZS. f. Phvs. 60,

156-167 (1930); 64, 1-21 (1930); 67, 194-206 (1931); 72, 528-541 (1931).

With F. Heidenreich: Natnrwiss. 21, 516-517 (1933).

E. Steudel: ZS.f. Phys. T7, 139-156 (1932).

Additional early references given at the end of Transmutation.

Transmutation with emission of neutrons:

G. Bernardini: ZS.f. Phys. 85, 555-558 (1933).
J. Chadwick: Proc. Roy. Soc. A142, 1-25 (1933).
N. Feather: Proc. Roy. Soc. A142, 689-714 (1933).

F. Joliot & I. Curie: /. de Phys. (7) 4, 278-286 (1933).

G. Kirsch & W. Slonek: Natnrwiss. 21, 62 (1933).
F. Rasetti: ZS.f. Phys. 78, 165-168 (1932).

Transmutation with emission of positive electrons:

1. Curie & F. Joliot: /. de Phys. (7) 4, 494-500 (1933).

Transmutation by Neutrons

N. Feather: Proc. Roy. Soc. A136, 703-727 (1932); 142, 689-709 (1933).

W. D. Harkins, D. M. Gans & H. W. Newson: Phys. Rev. (2) 44, 529-537 (1933).

Letters and abstracts by W. D. Harkins, D. M. Gans, H. W. Newson: Phvs.

Rev. (2) 43, 208, 362, 584, 1055 (1933); 44, 236, 310, 945 (1933).
F. N. D. Kurie: Phys. Rev. (2) 43, 771 (1933).

Abstracts of Technical Articles from Bell System Sources

Attenuation of Overland Radio Transmission in the Frequency Range
1.5 to 3.5 Megacycles per Second.^ C. N. Anderson. Data on the
effect of land upon radio transmission have been obtained during the
past few years in connection with various site surveys. These data
are for the general frequency range 1.5 to 3.5 megacycles per second
and for \arious combinations of overwater and overland transmission
as well as entirely overland. The generalizations in this paper are
chiefly in the form of curves which enable one to make approximations
of field strengths to be expected under the conditions noted above.
The relation of these data to transmission in the broadcast frequency
range is shown, and frorti the over-all picture, curves are developed
which enable field strength estimates to be made for overland trans-
mission in the extended frequency range.

The Radio Patrol System of the City of New York."^ F. VV. Cunning-
ham and T. W. Rochester. The application of radiotelephony to
municipal police work in New York City is described from the organ-
ization, viewpoint. Brief references are made to historical backgrounds
and description of apparatus, and the steps taken to select a receiver
suitable for local conditions are outlined. The method of controlling
the patrol force by radio is described at some length with examples,
and a summary of results during the first year is given to show the
value of this means of communication to police work.

Electrical Disturbances Apparently of Extraterrestrial Origin.^ Karl
G. Jansky. Electromagnetic waves of an unknown origin were
detected during a series of experiments on atmospherics at high
frequencies. Directional records have been taken of these waves for
a period of over a year. The data obtained from these records show
that the horizontal component of the direction of arrival changes
approximately 360 degrees in about 24 hours in a manner that is
accounted for by the daily rotation of the earth. Furthermore the
time at which these waves are a maximum and the direction from
which they come at that time changes gradually throughout the year
in a way that is accounted for by the rotation of the earth about the

^Proc. I. R. E., October, 1933.
^Proc. I. R. E., September, 1933.
^Proc. I. R. E., October, 1933.

159

160 BELL SYSTEM TECHNICAL JOURNAL

sun. These facts lead to the conclusion that the direction of arrival
of these waves is fixed in space; i.e., that the waves come from some
source outside the solar system. Although the right ascension of this
source can be determined from the data with considerable accuracy,
the error not being greater than ± 7.5 degrees, the limitations of the
apparatus and the errors that might be caused by the ionized layers
of the earth's atmosphere and by attenuation of the waves in passing
over the surface of the earth are such that the declination of the
source can be determined only approximately. Thus the value
obtained might be in error by as much as ± 30 degrees.

The data give for the coordinates of the region from which the
waves seem to come a right ascension of 18 hours and a declination of
— 10 degrees.

A Precision, High Power Metallo graphic Apparatus.'^ Francis F.
Lucas. In 1927 the design of an advanced type of metallographic
apparatus became of interest. Preliminary designs were prepared and
discussed at a conference in Jena, Germany, with the scientific staff
of Carl Zeiss. The Zeiss works was commissioned to construct the
apparatus. The work was directed by Professor A. Kohler, an out-
standing authority on the optics of the microscope, head of the mikro-
department of the Zeiss works, and Professor Walter Bauersfeld, a
director of the Zeiss Foundation and inventor of the Planetarium.

In this paper the author discusses the considerations which led to
the design and describes the construction of the apparatus. It is the
largest and the most powerful metallurgical microscope ever con-
structed. Capable of yielding crisp, brilliant images at magnifications
of 4000 to 6000 diameters, the design required great mechanical
stability, freedom from creep, absolute freedom from outside dis-
turbances, the means to illuminate the specimen with light of any
selected wave-length or group of wave-lengths within the visible
spectrum and the highest order of achievement in optical equipment.

^ Published in abridged form in Melal Progress, October, 1933.

Contributors to this Issue

H. S. Black, B.S. in Electrical Engineering, Worcester Polytechnic
Institute, 1921. Western Electric Company, Engineering Depart-
ment, 1921-25; Bell Telephone Laboratories, 1925-. Mr. Black's
work has had to do with the development of carrier telephone systems.

Arthur G. Chapman, E.E., University of Minnesota, 1911. Gen-
eral Electric Company, 1911-13. American Telephone and Telegraph
Company, Engineering Department, 1913-19, and Department of
Development and Research, 1919-. Mr. Chapman is in charge of a
group engaged in developing methods for reducing crosstalk between
communication circuits, both open wire and cable, and evaluating
effects of crosstalk on telephone and other services.

Karl K. Darrow, B.S., University of Chicago, 1911; University
of Paris, 1911-12; University of Berhn, 1912; Ph.D., University of
Chicago, 1917. Western Electric Company, 1917-25; Bell Telephone
Laboratories, 1925-. Dr. Darrow has been engaged largely in writing
on various fields of physics and the allied sciences.

Frederick B. Llewellyn, M.E., Stevens Institute of Technology,
1922 ; Ph.D., Columbia University, 1928. Western Electric Ccvmpany,
1923-25; Bell Telephone Laboratories, 1925-. Dr. Llewellyn has been
engaged in the investigation of special problems connected with radio
and vacuum tubes.

161

VOLUME Xm APRIL, 1934 NUMBER 2

THE BELL SYSTEM

TECHNICAL JOURNAL

DEVOTED TO THE SCIENTinC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION

The Carbon Microphone: An Account of Some Re-
searches Bearmg on Its Action — F. S. Gaucher 163

Open-Wire Crosstalk — A. G. Chapman 195

Symposium on Wire Transmission of Symphonic Music
and Its Reproduction in Auditory Perspective :

Basic Requirements — Harvey Fletcher .... 239

Physical Factors— 7. C. Steinberg and W. B. Snow 245

Loud Speakers and Microphones — E. C. Wente and
A. L. Thuras 259

Amplifiers — f. O. Scriven 278

Transmission Lines — H. A, Affel^ R. W. Chesnut
and R. H. Mills 285

System Adaptation — E. H. Bedell and Iden Kerney 301

Abstracts of Technical Papers 309

Contributors to this Issue 313

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

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The Bell System Technical Journal

April, 1934

The Carbon Microphone : An Account of Some Researches
Bearing on Its Action *

By F. S. GOUCHER

A great variety of speculations in regard to the physics of microphonic
action has arisen because of the complexity of behavior when current passes
through a so-called "loose contact" which forms the essential element in
a carbon microphone. Technical difficulties arising from the minuteness
of the contact forces and movements between contacts when in a sensitive
microphonic state have retarded the establishment of a quantitative theory.

Recent studies of carbon contacts have led to a satisfactory picture of
the nature of such contacts and their mode of operation when strained,
both from the elastic and the electrical point of view. The surfaces of the
carbon particles are microscopically rough and when two such surfaces are
brought together under the action of compressional forces, both the number
of hills in intimate contact and the contact area between hills vary through
deformations which are primarily elastic. Changes in electrical resistance
under strain are consistent with the assumption that current passes through
the regions in intimate contact.

Introduction

FEW electrical devices are as widely used as the "carbon micro-
phone" and few have given rise to as much speculation in regard
to their mode of action. That the problem has proved elusive is
shown by the fact that in Bell Telephone Laboratories it has been
regarded as perennial. However, recent researches have thrown a
considerable amount of light upon it and it therefore seems fitting to
bring before you this evening a brief survey of the subject and an
account of some of the latest experimental work.

The widespread use of the "carbon microphone "—it is employed
almost exclusively throughout the world in commercial telephone
service — is due primarily to its unique property of being its own
amplifier. In converting acoustical into electrical waves, it magnifies
the energy about one thousandfold. Other microphones, such as the
condenser or electromagnetic type, are unable to do this and so require
separate amplifiers when used in practice. For this reason, it seems
unlikely that the carbon microphone will be supplanted in the near
future for at least the great bulk of telephone work.

* Presented before the Franklin Institute, March 2, 1933. Published in the
Journal of the Franklin Institute, April, 1934.

163

164

BELL SYSTEM TECHNICAL JOURNAL

The essential element of this device is what has come to be called
the "loose contact" — or, as its name implies, a contact between two
conductive solids, metals as well as carbons, held together with small
forces. The ability of "loose contacts" to transmit speech was
discovered independently by Emile Berliner in this country and
Professor D. E. Hughes in England. Following Hughes* discovery,
Mr. Spottiswood, the president of the British Association in 1878,
described it thus: "The microphone affords another instance of the
unexpected value of minute variations — in this case, electric currents;
and it is remarkable that the gist of the instrument seems to be in
obtaining and perfecting that which electricians have hitherto most
scrupulously avoided, viz., 'loose contacts.'" Hughes applied the
word "microphone" to his instrument because of its remarkable
"ability to magnify weak sounds." The word itself is a revival of a
term first introduced by Wheatstone in 1827 for a purely acoustical
device developed to amplify weak sounds. Although originally con-
fined to the "loose contact" type of instrument, the term microphone
has more recently been used — particularly in broadcast, public
address, and sound picture work — for any device which converts
sound into corresponding electric currents.

Evolution of the Carbon Microphone
The story of the development of the "loose contact" type of micro-
phone is a fascinating one and, although it is beyond the scope of this

LINE

Fig. 1— Sketch, illustrating Bell's conception of the telephone, used in his first patent

application of 1876.

paper, 1 I should like to refer briefly to a few of the stages in the evo-
lution of the present day instrument. You will recall that Bell's
original telephone (Fig. 1) was electromagnetic in principle and acted

' I'or a more complete account see paper by H. A. Frederick, "The Development
of the Microi)hone," Bell Telephone Quarterly, July, 1931.

THE CARBON MICROPHONE

165

both as a transmitter and as a receiver. It was, however, very
Inefficient and Bell himself suggested that some other principle such
as that of variation of electrical resistance might overcome the
difficulty. He therefore devised the liquid transmitter In which a
small platinum wire (Fig. 2), attached to a drumhead of gold-beaters

Fig. 2 — Bell's liquid transmitter.

skin, is dipped into a small quantity of acidulated water in a con-
ducting cup. The extent of the area of contact between the liquid
and the wire is altered by the motion of the latter, thus altering the
resistance in a continuous manner. It was with this instrument that
the first complete sentence, "Mr. Watson come here — I want you,"
was successfully transmitted on March 10, 1876. This achievement

166

BELL SYSTEM TECHNICAL JOURNAL

stimulated others to work on the problem of a variable resistance
element and many new devices appeared in the next few years, the
most sensitive of which utilized a single loose contact, carbon in one
form or another being used as the contact material.

Fig. 3-

-Berliner's first single contact microphone, invented in 1877, employing a
metal-to-metal contact.

Figure 3 shows Berliner's first successful model consisting essentially
of a metal contact pressed against a metal diaphragm. This was
developed later into a carbon-to-carbon contact along the same
lines (Fig. 4).

Hughes, too, used metal in his first successful attempt at trans-
mitting sounds. Only three ordinary nails were required to demon-
strate the great sensitivity of loose contacts to acoustical vibrations
(Fig. 5). Hughes later developed the pencil type of microphone
(Fig. 6) in which carbon was used. It was the forerunner of many
practical devices developed along this line.

More rugged, reliable and permanent than either of these types was
the Blake transmitter shown in Fig. 7. It utilized a metal-to-carbon
contact and it owed its success to the mechanical control of the
contact pressure. This instrument was used for many years by the
Bell System.

Then came the Hunnings or the first of the granular carbon micro-

THE CARBON MICROPHONE

167

phones (Fig. 8), the immediate ancestor of the granular carbon type
used today. Hunnings used powdered "engine coke." It carried

Fig. 4 — Carbon-to-carbon single contact transmitter brought out in 1879 by Berliner.

more current than the Blake transmitter but it was liable to "pack"
and become insensitive.

This difficulty was overcome in the design invented by White in
1890, called the solid back type (Fig. 9). Millions of these are used
today in the ordinary desk-stand instrument. In this, carbon granules

Fig. 5 — Nail contacts used by Professor Hughes in 1878 to demonstrate their micro-
phonic properties.

168

BELL SYSTEM TECHNICAL JOURNAL

Pig 6 — Carbon pencil type microphone, mounted on a sounding board, demonstrated

by Hughes in 1878.

Fig. 7— The Blake transmitter using a platinum contact pressed against a carbon

block.

THE CARBON MICROPHONE

169

are compressed between two polished carbon electrodes which are
immersed in the granular mass in such a way that the particles have
more freedom of movement than in the Hunnings instrument. This
relieves excess pressure without undue packing.

Fig. 8 — Commercial model of the early Hunnings transmitter in which granular

material was first used.

In Fig. 10 we have a cross-sectional view of a modern handset
transmitter. This instrument, which is designed to operate in a
wide variety of positions, follows the Runnings' type in that the
granular mass rests against the diaphragm but it differs from it in
that the diaphragm does not act as an electrode. Both electrodes,
separated by an insulating barrier, form part of the containing walls
of the cell holding the carbon. This is the type which has recently
been studied in detail and of which a two dimensional model is shown
in Fig. 26.

The carbon used in these instruments is made by a heat treatment
of anthracite coal. The particles are about 0.01 inch in size and
when magnified they look just like lumps of coal taken from the
domestic pile (Fig. 11).

Speculations of the Early Inventors
Part of the difficulty in elucidating the microphonic action of the
"loose contact" arises because so many effects can be observed or are

170 BELL SYSTEM TECHNICAL JOURNAL

associated with the action that it is hard to determine which of them
is essential. It is therefore not surprising that there was great diver-
sity of opinion amongst the early inventors.

Fig. 9 — The solid back transmitter invented by White in 1890.

For instance, experiment shows that contacts tend to move apart
when in the act of transmitting sound. This led many, amongst
them Berliner, to hold the view that an air film is necessary for micro-
phonic action, that the current somehow passes through the film, and
that the variation of the current is due to the variation of the thickness
of the film. This view, however, was partly discredited by experi-
ments showing that the moving apart was probably due to a heating
of the contact through the passage of current and hence that it is
not a necessary accompaniment of microphonic action.

Again, when one listens through a receiver placed in a circuit con-
taining a "loose contact," noises are heard, especially when the

THE CARBON MICROPHONE

171

voltage across the contact or microphone is large. These noises are
irregular like frying or crackling. Also, if a contact be viewed under
a microscope, bright spots are sometimes seen. These facts have led
many to think that small arcs are always present and are responsible
for microphonic action. Hughes was very much inclined to this view.

Fig. 10 — Cross-section of the barrier type transmitter used in modern handset

instruments.

There were reasons for supposing that the heating of the contact is
a necessary factor in microphonic action. This point of view was
supported by Preece, who wrote in 1893, "Indeed there are many
phenomena such as hissing and humming that are clearly due to
what is known as the Trevelyan efifect, that is, the motion set up by
expansion and contraction of bodies which are subjected to variation in
temperature. This at least tends to favor the heat hypothesis as
does also the fact that with continuous use some transmitters become
essentially warm."

172 BELL SYSTEM TECHNICAL JOURNAL

Another view was that microphonic action arises from change in
resistivity of the solid carbon resulting from strain. This view was
held by Edison who doubtless believed it because of the success of
his microphone which was designed with the object of applying
pressure variation to a solid carbon block. It failed of general

Fig. 11 — Carbon granules made from anthracite coal (X 15).

acceptance because the effect of pressure on resistance, as shown by
experiment, seemed definitely to be too small. It was generally con-
sidered that the Edison instrument was in fact a "loose contact"
although Edison himself did not realize it.

Others of the early inventors considered the contact area to be the
essential element — that is to say, the extent of surface or the number
of molecules involved in intimate contact. As Professor Sylvanus
Thompson expressed it in 1883, "An extremely minute motion of
approach or recession may suffice to alter very greatly the number of
molecules in contact. . . . Just as in a system of electric lamps in
parallel arc the resistance of the system increases when the number of
lamps is diminished and diminishes when the number of lamps con-
necting the parallel mains is increased, so it is with the molecules at
the two surfaces of contact."

THE CARBON MICROPHONE 173

Recent Theories

The first attempt at a quantitative theory of microphonic action
was made by Professor P. O. Pedersen in 1916.^ He assumed that
microphonic action is due to the variation of the contact area arising
from the elastic deformation of the contact material by pressure.
Considering the case of two elastic conducting spheres brought into
contact, Pedersen assumed that the resistance is made up of two
parts; viz., (1) the resistance of a conducting film having a specific
resistivity differing from bulk carbon and independent of pressure,
and (2) the so-called "spreading resistance" or that which is caused
by the concentration of the current flow within the region of the
contact area and which would exist independently of any film.

This theory results in a quantitative expression ^ for the dependence
of the contact resistance on the force holding the contacts together.
Pedersen tested it by experiments on carbon spheres and found
reasonable agreement over a wide range of force. However a very
similar expression can be obtained without postulating the existence
of the high resistance film. We have merely to suppose that contact
does not take place over the whole contact area owing to surface
roughness (the existence of which can be observed under a microscope,
especially in the case of carbon).

Dr. F. Gray of Bell Telephone Laboratories worked out an ex-
pression ^ based on this assumption which was so nearly like Pedersen's
that it was difficult to discriminate between them experimentally.
He assumed both that the number of microscopic hills in electrical
contact increases as the contact force is increased and that the re-
sistance per hill varies in accordance with the theory of spreading
resistance as assumed by Pedersen. His equation was found to fit
experimental curves remarkably well for contact forces which are
relatively larger than those holding the granules together in a micro-
phone. In the range of smaller forces, however, marked departures
from theory were found, the measured value of resistance decreasing
too rapidly with an increase of force. Although these departures
were believed to be due at least in part to a plastic deformation of
the contact material, it appeared possible that other factors come into
play and may even be dominant in this region of small contact forces.

For instance, it had been demonstrated that adsorbed films of air
are capable of producing a marked increase in the resistance of granular
carbon contacts. This revived the air film theory as a possibility
under the condition of small contact forces.

2 The Electrician, Jan. 28-Feb. 4, 1916.

37? = AF-^'^ + BF-^i\

*R = AF-^l^ + 5/^1/3 {Phys. Rev., 36, 375, 1930).

174 BELL SYSTEM TECHNICAL JOURNAL

Again there is a marked decrease in the resistance of granular
carbon contacts with increase in voltage which had not been satis-
factorily explained. This fact suggested amongst other possibilities
that the conduction process may involve the passage of electrons
across gaps of molecular dimensions in the manner of a cold point
discharge. Field gradients of sufficient magnitude to extract electrons
from a solid must exist in these gaps with only a fraction of a volt
across the contacts. If this is the main process by which current
passes between contacts, microphonic action might well be associated
with a variation of the gap dimensions under strain.

Again recent work on the theoretical strength of solids had led to
experimental results showing that under certain conditions solids
may, without fracture, be subjected to strains greatly exceeding those
heretofore obtained. This suggested the possibility that the micro-
phonic effect of contacts might after all be associated with the straining
of small junctions welded under pressure and current.

In view of the speculative nature of the situation it was clear that
a new experimental attack on the problem was necessary. We have
been making such an attack during the last few years and I now turn
attention to some of the experimental results and the main conclusions
to be drawn from them.

Recent Experimental Work
Statement of the Problem

Since the essential element in the carbon microphone is the so-
called "loose contact," the first and most fundamental step toward
the understanding of the physics of microphones is the solution of
the problem of the "loose contact" when in its sensitive or microphonic
state.

Measurements on microphones such as the handset have enabled us
to specify pretty accurately the conditions under which any two
granules within the structure operate when the microphone is trans-
mitting speech or sound.

In addition to the voltage, which is limited to one volt per contact,
these conditions may be stated briefly either in terms of contact
forces or in terms of movements between centres of granules. When
you realize how small these are — particularly the movements between
centres of granules — you will, I think, not be surprised that the solu-
tion of the problem of the "loose contact" has been so long delayed.

For the condition of reasonably loud speech the diaphragm motion
is about

1 X 10 =^ cm.,

THE CARBON MICROPHONE 175

which is just on the limit of resolution of the highest-power micro-
scopes. It follows from a consideration of the number of granules in
series that the movement between centres of granules w^ould not be
greater than 1/lOth of this, viz.,

1 X 10-« cm., '

which is in the submicroscopic range. We must, therefore, be able
to control and measure movements at least as small as 10"'^ cm.; not
an easy thing to do with a "loose contact."

The contact forces are on the average somewhat less than 10 dynes
when the aggregate is in the unagitated state. In the presence of
acoustic waves, variable forces of several dynes are superimposed on
these fixed forces. The variable forces are smaller than the fixed
forces, so that the granules will on the average remain in contact
throughout any reversible cycle. We have reason to believe that 10
dynes is about the maximum force which is attained at any one
contact during a stress cycle. We must therefore be able to control
contact forces within the range 1 to 10 dynes.

Apparatus and technique have now been developed for studying
single contacts within the prescribed range of forces and displace-
ments, and significant measurements have been made which I will
now endeavor to describe to you somewhat in detail.

Single Contact Studies

Figure 12 shows the construction of one of the contact tubes used
in this study.

Its essential features are shown diagrammatically in Fig. 13. The
contact pieces C\ and Ci are fastened respectively to a movable base
M and to the lower end of a helical spring made of fused quartz. The
base is supported from a fixed frame by two vertical platinum wires
P and two stretched springs as shown. The lower contact piece is
moved by heating or cooling the platinum wires through the passage
of current. In this way the contacts may be made or broken and
any desired contact force applied, the measure of the force being the
compression of the helical spring. The temperature of the contact
is varied by surrounding the contact region with a metal cylinder 5
which may be heated by means of radiation from a coil of platinum
wire H, the temperature within the cylinder being measured by means
of a thermocouple placed near the contacts.

In practice the upper contact piece consists of a single granule
fastened to the end of a platinum wire and the lower contact piece
consists of a number of granules attached to a horizontal metal plate;

176

BELL SYSTEM TECHNICAL JOURNAL

Fig. 12 — Device for controlling force and temperature used in the study of single

contacts.

THE CARBON MICROPHONE

177

in this way a variety of contacts can be studied with the same tube.
A small hole in the metal cylinder permits of direct observation of
the contacts during measurement. Figure 14 shows how the apparatus

Fig. 13 — Diagrammatic view of single contact device shown in Fig. 12.

was mounted in an iron cylinder on a damped suspension to protect it
from acoustical and mechanical disturbance. The two microscopes
were used to observe the compression of the silica spring.

We first studied the effect of voltage and temperature on contacts
held together with constant forces. Reversible characteristics could
in all cases be obtained for voltages up to 1 volt and for temperatures
up to about 80° C.

Typical characteristics are shown on Fig. 15 in which the contact
forces were of the order of 1 dyne. On the left are plotted the re-
sistance-voltage characteristics and on the right the resistance-
temperature characteristics. All of the variables are plotted for con-
venience on logarithmic scales.

178

BELL SYSTEM TECHNICAL JOURNAL

The curves /, // and /// illustrate the fact that Ohm's law is found
to hold for all contacts up to about 0.1 volt and that above these
values the contact resistance decreases with increase of voltage.

Fig.

14 — The single contact device is mounted in a heavy container on a spring
suspension to minimize acoustic and mechanical disturbance.

The fractional decrease in resistance with voltage above 0.1 volt is
independent of the contact resistance and whether or not the measure-
ments are made in air or vacuum.

In curves /', //' and ///' we have changed the voltage scale of the
curves /, // and III to a temperature scale in accordance with the
relation,

T = 7^0 + 40 71

THE CARBON MICROPHONE

179

This relation has a theoretical basis in the Joule heating of the
contacts due to the passage of current and contains the assumption
of a value of Wiedemann Franz ratios characteristic of solid carbon.*

IXIO'

1X10-

1X10

n AIR

-^^

m VACUUM

.^_^

• BY CHANGING TEMPERATURE

o CALCULATED FROM T = T^ + 40 V
V

0.001

0.01 0.1

VOLTS

1.0 10

50 100

TEMPERATURE IN°C

Fig. 15 — Characteristics showing the effect of voltage and temperature on contact

resistance.

These curves have substantially the same slope as A, which is a
characteristic measured by heating a contact in the furnace, the con-
tact voltage being sufficiently small to avoid appreciable heating of
the contact due to this cause, and also with B, which was obtained with
a solid carbon wire produced in a manner to simulate closely micro-
phone carbon. We are able to conclude from measurements such as
these that the nature of the conducting portions of contacts is that of
solid carbon both for air and vacuum and that the departures from
Ohm's law — at least up to 1 volt — are due to the Joule heating of the
contacts.

From measurements similar to these in which we show that the
admission of air has no effect on the temperature coefficient of re-
sistance— although it produces a marked increase in the resistance at
any particular temperature — we are also able to conclude that the
presence of adsorbed air does not alter the nature of the conducting
portions of the contacts but merely limits their areas.

* This theory, based on earlier work of Kohlrausch, was worked out in useful
form independently in Bell Telephone Laboratories (unpublished work) and by R.

Holm {Zeit. Tech. Phys., 3, 1922). It gives the approximate relation, const, j^ . ,

as the increase in temperature above room temperature, V being the contact volt-
age, and Ko/ffo the Wiedemann Franz ratio for the contact material.

180

BELL SYSTEM TECHNICAL JOURNAL

Turning now to the effect of contact force on contact resistance:
we see (Fig. 16) that large and approximately reversible resistance
changes are produced as the force is varied repeatedly between fixed
limits. This shows that the effect is in the main elastic, though the

320

300
280

260

240

220
<n
a. 200

< 180

t 160

a. 140

120

100
60

X 10"

€^

0 12 3 4 5 6 7

FORCE IN DYNES

Fig. 16 — Typical current-force cycle obtained with a single contact.

existence of a narrow loop indicates a small plastic or irreversible
movement as a secondary effect.

We have^ thus established that the current is conducted through!
solid carbon and that the deformations are mainly elastic. These!
facts give strong support to the "elastic theory" of "loose contacts,"
i.e., the hypothesis that the change of resistance takes place becausel
of a change in contact area under pressure. An extensive study off
the resistance-force characteristics gave results which could not be

THE CARBON MICROPHONE 181

simply interpreted (just as Gray had found) and, because of the
possibility that unknown cohesional or frictional forces were involved,
the work was extended by a study of resistance-displacement charac-
teristics. Through a comparison of the two sets of data we were
led to the conclusions that the stress-strain characteristics are not so
simple as those assumed in Pedersen's or Gray's analysis and, there-
fore, that a study of the elastic behavior of contacts offered the most
promising line of attack on the problem.

Figure 17 shows the mechanical system developed for this purpose.
With it known forces can be applied to a contact element and at the
same time its movement can be measured.

The contact is made between a carbon granule and a polished
carbon plate, the granule being attached to the end of a rod R sus-
pended by springs 5' from a fixed frame and the plate being attached
to the end of a micrometer screw M2 capable of giving to it a transla-
tional motion without rotation.

The force is applied to the granule electrostatically by means of
voltage applied between the condenser plates C2, one of which is
attached to the rod R and the other to the micrometer screw. This
is in principle the attracted disc electrometer of Kelvin and it is
capable of applying forces up to 15 dynes without using voltages
greater than 200.

The motion of the granule with respect to the carbon plate is
measured electrically through the variation of capacity of the con-
denser Ci, of which one plate is attached to the other end of the rod R.
Ci forms part of an oscillating circuit of natural frequency Wo (about
2000 kc.) which is coupled to a wave-meter circuit adjusted for oscilla-
tion at a frequency «i slightly different from n^. Changes in the
frequency arising from the changes in capacity C\ alter the energy
picked up by the wave-meter circuit and this energy, which is recorded
by means of a galvanometer, serves as a measure of the change of
capacity or motion of the rod R. With this arrangement it is possible
to measure motions as small as 1 X 10"'^ cm. and under the best con-
ditions as small as 1 X 10~^ cm. It is necessary to have good damping,
which is obtained by means of immersing the drum D in polymerized
castor oil. The accessory spring ^2 is used merely for calibrating
purposes.

Figure 18 shows the appearance of the apparatus as set up for
measurement. The condenser is contained in the lower housing at
the left, the wave-meter in the upper housing. The whole apparatus
including the galvanometer is supported on a delicate spring suspension
within a second large lead container, the frame of which just appears
at the edge of the photograph and which is also supported by springs.

182

BELL SYSTEM TECHNICAL JOURNAL

THE CARBON MICROPHONE

183

Figure 19 shows the appearance of the complete setup with the
cover on the outside container. This begins to compete with cosmic
ray apparatus from the point of view of the amount of lead involved,
the outer container weighing about 600 lbs. Port-holes — one of

Fig. 18 — Mechanical system and associated electrical apparatus as set up for single

contact study.

which appears on the near end of the box — permit adjustments to be
made on the apparatus within, thus eliminating the necessity for re-
moving the large outer cover which, as you may surmise from the
number of handles, requires the combined efforts of two men to

184

BELL SYSTEM TECHNICAL JOURNAL

remove it. All of this protection is, of course, to shield the apparatus
from mechanical vibrations and acoustic disturbances.

Fig. 19— Exterior view of complete experimental arrangement.

With this apparatus we investigated the variations in displacement
and resistance when the forces are varied cyclically between fixed
limits. Measurements on a large number of contacts are summarized
in curves. Figs. 20 and 21. The cyclic characteristics, though some-
what irregular and having the form of narrow loops, approximate
straight lines when the variables are plotted on logarithmic scales.
Only one complete characteristic is shown in each set of curves, other
typical measurements being represented by dotted straight lines
joining the end points of their respective cycles. The full line in
each figure represents the cycle of a typical contact, obtained by
averaging, over the range in which the difTerence between the maximum
and minimum force limits or maximum and minimum displacement
limits is relatively large, in which case the slope is apparently con-
stant.

If we let N" and N represent the slopes of the typical force-displace-

THE CARBON MICROPHONE

185

ment and resistance-force characteristics and if F, D and R be the
contact force, the contact displacement and the contact resistance,

•

}

t
1

/ /

1
1

f
1

1
1

t ,

' i

1 1

^ J

1
1

// /

r/ /

/
/

' Jl

r /

/
/

i
I

'/ 1

/
/
/

F = CONSTANT X d"^
AVERAGE n"= 3.1

DISPLACEMENT IN CENTIMETERS X 10"

Fig. 20 — Typical force-displacement characteristics of carbon granules pressed
against a polished carbon plate.

respectively, we may express our results by the approximate relations:

F = const. -i)^",
R = const. -T^-^.

(1)
(2)

The values N" and N are not, however, independent of the force or
displacement limits when these limits are relatively small. In Fig. 22
we have plotted values of N" and N as functions of the difference
between the maximum and minimum displacement (AD). We see
that for relatively large values of AD, N" and N approach the limiting
values 3.1 and 0.47, respectively, but for smaller values of AD, N"

186

BELL SYSTEM TECHNICAL JOURNAL

1000
900
800

700
600

500

2 400

Z 300
<

200

100

^k,^

o,,^^

—

■ — ■

**«.

•^^

""^

"■ «««.

'^^

■ — ,

•^^

^-L

^>^,

»»

■>>v.

■^

'">*

'"x.

•>•

^v.

-V.

■>

■"^^

•

■^^

"•v^ ,

^■s.

^"s.

^s. ^

■^

R = CONSTANT X F""^
AVERAGE N = 0.47

■v.

"-.

3 4

FORCE IN DYNES

7 8 9 10

Fig. 21 — Typical resistance-force characteristics of carbon granules pressed against

a polished carbon plate.

\
\

— -^^ — o

/
/

0 2 4 6 8 10 12 14 16 18 20 22

AD IN CM X 10"^

Fig. 22 — Effect of the extent of contact motion (A£>j on A^and A^". (Average values.)

THE CARBON MICROPHONE 187

becomes greater than and N less than its Hmiting value. The limiting
value of N" is greater than that which would be obtained through the
contact of hemispherical surfaces and represents a more rapid stiffening
of the contact with compression.

We will first give our attention to the limiting value of N".

A consideration of the nature of contact surfaces as revealed by the
microscope furnished the clue to the interpretation of our results.
A typical surface is shown in the photomicrograph (Fig. 23). Evi-

Fig. 23 — Photomicrograph of the surface of a carbon granule (X 240U).

dently it is very hilly, the hills being much the same size and height.
The magnification (X 2400) is such that the small white circle has a
diameter of 8 X 10~^ cm. and it is clear that the circle encloses several
hills.

From the theory of elasticity we may deduce that if two hemi-
spherical hills of carbon having a radius of the order 1 X 10~^ cm. are
brought together with forces of the order of 1 dyne the maximum
stresses will probably not exceed the elastic limit of carbon and hence
that the hills will deform elastically. The motion involved in such a
deformation will be of the order of 1 X 10~® cm. and if other hills are
encountered, as is most probable with such a movement, the stresses
will be shared and hence the stress per hill reduced. According to
this view forces larger than one dyne can be applied without exceeding
the elastic limit merely by virtue of the distribution of the hills which
will come in to share the stresses. Furthermore, such a contact will

188

BELL SYSTEM TECHNICAL JOURNAL

stiffen up more rapidly with compressional displacement than will a
contact made on a single hill. This concept of a loose contact,
therefore, seemed to offer possibilities in the way of an adequate ex-
planation of the experimental results.

At first the problem seemed too complex for mathematical analysis
and a study of the elastic behavior of contact surfaces having various
arrangements of little hemispherical hills was made with the aid of
large scale rubber models. Quarter inch rubber balls were cut in
half for this purpose and arranged on bases of suitable material and
shape.

0.3
0.2

0.1

I - SMOOTH

SPHERE

UNIFORM DISTRIBUTION
m- HEMISPHERES ON PLANE,

PROBABILITY DISTRIBUTION

m. 1

1 /

0.05

0.04

0.03
0.02

0.01

SL<

DPE

: 1

I'f

/ A. 2

(2.b

0.005
0.004

0.003
o.no?

0.001

0.05

0.1

0.005 0.01

DISPLACEMENT IN CENTIMETERS

Fig. 24 — Stress-strain characteristics obtained with contact surfaces made of rubber.

In Fig. 24 we have plotted the force-displacement characteristics
of three different surfaces: /, that of a single smooth hemisphere;
//, that of small hemispheres of equal height evenly distributed on a
portion of a large 32 inch sphere made also with rubber; and ///,

THE CARBON MICROPHONE 189

that of hemispherical surfaces of random height fastened to a flat
plate, about 100 hemispheres being used.

We see from the slopes of these curves that the model made with
hills of random height on a flat plate behaves most like the actual
contacts, the slopes of the corresponding curves being 3.2 and 3.1,
respectively. This arrangement is also the one which most nearly
represents the carbon surfaces as viewed under the microscope. Here
the hills have various heights and the radius of the underlying base
(0.015 cm.) is so much larger (1000 fold) than that of the average hill
that within the region of the contact area the surface of the former
may be regarded as plane.

The slope of curve / is in accord with a formula derived from the
theory of elasticity by Hertz connecting the force F pressing together
two elastic spheres and the movement D between the centres of the
spheres:

F = const. i)3/2 (3)

The constant includes such factors as the elastic moduli of the contact
materials and the radii of the spheres and need not concern us here.
The case of a sphere pressed against a flat plate, as in our experiments,
is a particular case of this general equation, the constant only being
affected.®

The slopes of curves // and III are also in accord with theory, as
we shall see, when one makes the simple assumption that the elastic
deformation is confined to such a small region near the contact in each
hill that the underlying base is not appreciably deformed. This
assumption was tested in the case of the model having the spherical
distribution of hemispheres by changing the stiffness of the rubber
used in the underlying sphere. No effect was produced on the stress-
strain characteristic (curve //). We may therefore consider that the
elastic reactions produced in each hill are independent of each other
and that the base is not deformed, so that with a given distribution
of hills it becomes a simple matter to calculate their combined effect
over a given compressional range. We may represent the conditions
essential for our calculation by the diagram, Fig. 25, in which A
represents the plane surface of the smooth contact element just making
contact with the highest hill of the rough contact element. Under
compression, A may be considered as moving in the direction of its
normal x, compressing B and, with increasing motion, coming into

* Formula (3) is known to hold accurately for values of D not greater than about
1 per cent, of the radius of the sphere (J. P. Andrews, Phys. Soc. Proc, Vol. 42, No.
236). This condition is fulfilled in the case of curve I but D is as great as 10 per cent.
of the radius in the case of a few of the hills involved in the maximum compression
shown in curves // and /// (Fig. 24).

190

BELL SYSTEM TECHNICAL JOURNAL

contact with other hills C and compressing them according to equation
(3). The position of C is conveniently defined by its distance X from
the plane A.

Fig. 25 — Schematic representation of a rough surface used in mathematical analysis.

Any continuous distribution of hill positions, typified by C, which
would be encountered through a small compressional movement, may
be approximately represented by the expression.

Nx = const. .Y",

(4)

where iVx is defined as the number which multiplied by dx gives the
number of hills coming into contact with the plane when it moves
from X to .r + dx. The exponent « is a constant which for convenience
we may call the distribution constant.

For a total compression D the N^dx hills will be compressed an
amount D — x, and hence the total force of reaction F is given by

F = const

. rx"{D -

xyi^ix,

which integrates to the form,

F = const. Z)"+"^/2 = const. Z)^". (5)

The constant here includes a summation of the individual constants
of equation (3). It is clear that if the hills have different radii the
constant only will be affected, so that equation (5) may be regarded
as general in this respect.

THE CARBON MICROPHONE 191

For the case of uniform hills distributed on the surface of a sphere
it may be shown that equal numbers of hills will be added for equal
increments in x, in which case Nx = constJ From this it follows
that n = 0 and N" = 2.5 in agreement with the measured value,
curve 11.^ For N" = 3.2 as obtained with the hemispheres of random
height on a plane, curve ///, n would have the value 0.7. The
corresponding distribution function N^ would approximate to that of
the portion of an ordinary error curve near its maximum. A rough
determination of the distribution of heights amongst the small rubber
hemispheres showed in fact that they approximated closely to an
error curve and that the displacement range covered that portion of
the curve near the most probable height.

It would appear from this analysis that the elastic behavior of our
carbon contacts under conditions of relatively large strain is adequately
explained on the very simple assumption that the hills which we
observe under the microscope have a random distribution of heights
and behave like smooth spherical surfaces. We have, however, still
to account for the hysteresis and the large values of N" corresponding
to small values of AD as well as the values of N (Fig. 22).

It is unlikely that the hills which we observe under the microscope
are submicroscopically smooth, in which case we would expect a
small plastic movement in these secondary hills arising from overstrain.
We have direct evidence for this in the fact that contacts once estab-
lished— even without the passage of current — require relatively small
but finite forces to break them. Such junctions within the contact
region could well account for hysteresis and a stiffening up of the
contact in the region of small strains. Furthermore it is to be expected
that they might affect the resistance behavior to a much greater
extent than the elastic, and over a wider range of strain, since the
junctions — ^though too weak to affect appreciably the contact stiffness
— ^might well carry a relatively large proportion of current; in which
case the value of N would be smaller than that calculated on the
assumption of smooth spherical surfaces.

We will now derive an expression relating resistance and force for
the type of contact considered in the derivation of equation (5),
assuming smooth hills.

Classical theory ^ gives the following formula for the conductance

' This argument rests on the fact of geometry that if A is the area of contact
between a sphere of radius r and a plane, dA/dx = 2irr.

* This agreement between theory and experiment shows that the compression
of some of the hills by an amount in excess of 1 per cent, of their radii has not aflfected
the applicability of equation (3) to our problem.

" Riemann Weber.

It is here assumed that the mechanical and electrical areas of contact are coinci-
dent, which according to the ideas of wave mechanics may not be the case.

192 BELL SYSTEM TECHNICAL JOURNAL

l/r of the contact formed by compressing, by an amount D, a single
smooth conducting sphere against a flat conducting plate,

- = const. D'i\ (6)

It appears reasonable to assume that the hills which come into contact
with compression act independently of each other as regards con-
duction. The conductances may therefore be added and we may
write for the total conductance (l/i?) produced by a compression D
involving many hills:

1
R

= const.

/•%.

./()

'(^

- xyi-'dx,

ich

integrates

to the form,

1
R

const.

+3/2

ich

in combin;

ation

with (5)

gives

= const.

2«+3

7?2n+5

= const, f-'^.

(7)

Using the value of n consistent with equation (5) through the measured
value of TV", viz., n = 0.6, we get N = 0.68. The measured value
of A^ (0.47) is, as we have surmised, too small though it is of the right
order of magnitude.

We are, of course, investigating the factors which give rise to this
discrepancy as they will play an important part in any complete
theory of microphonic action, and we are extending our study to the
behavior of granular aggregates in simple cells and microphone
structures. We have shown that the value of iV in a simple cell com-
posed of parallel electrodes is quite consistent with our simple theory
for single contacts, which therefore indicates that the behavior of an
aggregate of contacts is determined by the behavior of the individual
contact. Furthermore, we have shown, through static measurements
on the handset instrument, that the granular aggregate within this
irregularly shaped structure behaves like the aggregate in a simple
cell. We are therefore confident that the behavior of the microphone
will be explained in terms of the behavior of the single contact.

The behavior of the two dimensional model of the handset micro-
phone (Fig. 26) is most convincing in this connection. Although this
model was set up originally to study the distribution of stresses in
this type of structure it has proved most useful in other phases of our
work. Quarter inch rubber balls represent the granular particles of

THE CARBON MICROPHONE

193

Fig. 26 — Model of handset transmitter cell.

525

0425

z
<

N.N

275

_ \

4 5 6 7 8

FORCE IN GRAMS

II 12

Fig. 27 — Resistance-force cycle obtained with transmitter model.

194

BELL SYSTEM TECHNICAL JOURNAL

the actual microphone and by coating these with a conducting layer
of graphite and lacquer we are able to make them behave electrically
as well as elastically in accordance with our simple theory. When
placed in the model the aggregate is compressed cyclically by means
of the piston which acts as a diaphragm, producing a change of re-
sistance in the current path around the insulating barrier. The
curves shown in Figs. 27 and 28 show typical resistance-force cycles,
obtained with the model and the actual instrument under conditions
wherein the reactive forces are mainly elastic. The similarity of
these characteristics is striking. The existence of the loops indicates
that the reactive forces are not entirely elastic and that the behavior
is modified by friction, as in the case of single contacts.

f) 95

Z 90

v ^

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

FORCE IN GRAMS

Fig. 28 — Resistance-force cycle obtained with a standard transmitter.

In conclusion it seems fair to say that our experiments on "loose
contacts" under conditions which are equivalent to those under which
they operate in actual microphones have given a satisfactory picture
of the essential nature of such contacts, and their mode of operation
when strained, both from the elastic and the electrical point of view.
The electrical current is carried through regions in intimate contact
and changes in resistance under strain are due both to a variation in
the number of microscopic hills which form the carbon surface and to
area changes at the junctions of these hills arising from their elastic
deformation in accordance with the well known laws of elasticity.

open- Wire Crosstalk *

By A. G. CHAPMAN

Effect of Constructional Irregularities

IF the cross-sectional dimensions of an open-wire line were exactly
the same at all points and if the transpositions were located at
exactly the theoretical points, the crosstalk could be reduced by huge
ratios by choosing a suitable transposition arrangement and interval
between the transposition poles.

Practically, however, the crosstalk reduction is limited by un-
avoidable irregularities in the spacing of the wires and of the trans-
position poles. There is no point in reducing the type unbalances by
transposition design beyond the point where the constructional
irregularities control the crosstalk.

Transposition Pole Spacing Irregularities

The following discussion covers the method of estimating the

crosstalk due to irregularities in the spacing of transposition poles and

the derivation of rules for limiting such irregularities. With practical

methods of locating transposition poles, the effect of the pole spacing

irregularities may ordinarily be calculated by considering only the

transverse crosstalk. Special conditions for which attention must be

paid to interaction crosstalk are discussed later. The simplest case,

that of transverse far-end crosstalk due to pole spacing irregularities,

will be discussed first.

A transposition section is divided into segments by transposition

poles which in practice vary in number from four to 128. Each

segment causes an element of crosstalk current at a circuit terminal

and this element is about proportional to the segment length. For

far-end crosstalk between similar circuits all these crosstalk current

elements would add almost directly if there were no transpositions.

The function of the transpositions is to reverse the phase of half the

current elements. The segments corresponding to the reversed current

elements may be called the minus segments. If the other half of the

* This is the second half of a paper which was begun in the January 1934 issue
of the Technical Journal, giving a comprehensive discussion of the fundamental
principles of crosstalk between open-wire circuits and their application to the trans-
position design theory and technique which have been developed over a period of
years.

195

196 BELL SYSTEM TECHNICAL JOURNAL

segments are called the plus segments, the far-end crosstalk is pro-
portional to the difference of the sum of the plus segments and the
sum of the minus segments. This difference may be called the
unbalanced length and the output-to-output far-end crosstalk is this
length multiplied by the far-end coefficient and by the frequency.

If the sum of the plus segments equals the sum of the minus seg-
ments, the unbalanced length will be zero. The poles of a line are
necessarily spaced somewhat irregularly but for a single circuit
combination the unbalanced length could be made very small by
carefully picking the transposition poles so as to keep the sums of the
plus and minus segments about equal. This procedure is impractical,
however, because many circuit combinations must be considered and
because necessary line changes would prevent the maintenance of very
low initial unbalanced lengths.

In practice, therefore, the segment lengths are allowed to deviate
in a chance fashion from the mean segment length. The unbalanced
length varies among the various circuit combinations depending on the
arrangement of the transpositions which determines the order in
which plus and minus segments occur. For any particular combina-
tion, the unbalanced length has a wide range of possible values and
its sign is equally likely to be plus or minus.

In any transposition section, the length of any segment may deviate
from the average segment length for that section. If the sum of the
squares of all the deviations in each transposition section is known,
the unbalanced length for a succession of transposition sections may
be estimated, that is, the chance of the total unbalanced length lying
in any range of values may be estimated.

Letting ^'i^ be the sum of the squares of the deviations for the first
transposition section, etc., and letting R be the r.m.s. of all the possible
values of the total unbalanced length in all the sections, the following
approximate relation may be written:

i?2 = 5^2 _^ 5,2 _^ . . . etc.

The chance of exceeding the value R may then be computed. For
example, there is about a one per cent chance that the total unbalanced
length will exceed 2.6R.

In making rules for locating transposition poles the first step is to
determine a value for R. For example, if consideration of tolerable
crosstalk coupling indicated that there should not be more than one
per cent chance that the total unbalanced length in a 100-mile line
would exceed one mile, then R, the r.m.s. of all possible values of the
total unbalanced length, should not exceed 1/2.6 miles. Since R is

OPEN-WIRE CROSSTALK 197

calculated from the values of 5" for the individual transposition
sections, a given permissible value of R may be obtained with various
sets of values of S. It seems reasonable to determine individual
values of 5 on the principle that a transposition section of length Lg
should have the same probability of exceeding a given unbalanced
length as any other section of the same length and that a section of
length 2Ls should have the same probability as two sections of length
Lg, etc. On this basis, the value of S- for any transposition section
should be proportional to the section length Ls. This leads to the
rule used in practice that for any transposition section 5^ should not
exceed kL^. If Ls and 5 are expressed in feet, a value of three for k
is found suitable for practical use. The choice of a value for k will
depend, of course, upon the cost of locating and maintaining trans-
position poles with various degrees of accuracy and upon the effect
on the crosstalk of varying the value of k.

The above rule permits a large deviation at one point in a trans-
position section if it is compensated by small deviations in the rest
of the segments. For example, with 128 segments and a mean
segment length of 260 feet, one long segment of 575 feet is permissible
if the rest of the segments are 258 feet. The expression for the total
unbalanced length in a succession of transposition sections assumed
that the deviations varied from segment to segment in a truly random
manner. The above example involves an unusual arrangement of
the deviations. When there are a number of transposition sections
in a line, such unusual arrangements of deviations in various sections
do not have much effect on the probability that the total unbalanced
length will exceed a given value.

The computation of near-end crosstalk due to pole spacing irregu-
larities is a more complicated problem since the crosstalk elements
resulting from the various segments vary in their magnitudes and
phase relations because the various segments involve different propa-
gation distances. It may be concluded, however, that the r.m.s.
value of the total unbalanced length in all the sections may be ex-
pressed as follows:

This differs from the expression for far-end crosstalk in that the
values of S"^ for the second and succeeding transposition sections are
multiplied by attenuation factors. The attenuation factor A^ cor-
responds to propagation through the first section to the second section
and back again. The other attenuation factors are similarly defined.
The above expression neglects attenuation within any particular

198 BELL SYSTEM TECHNICAL JOURNAL

transposition section since this is ordinarily small. It also assumes
that the rule for locating transposition poles, that is, that S^ should
not exceed kL2, is applied for lengths having only negligible attenuation.
In making estimates of R in connection with transposition design
work, it is assumed that all the segments are nominally the same
length, D, and that r is the r.m.s. value of the deviations of the seg-
ments. Since r^ equals S^ divided by the number of segments in
length La, r"^ should not exceed kD. B} may be expressed approxi-
mately in terms of r- as follows:

1 - €-^«^

i?2 = ^2

\ _ g-4aZ> >

where R and r are expressed in the same units, L is the length of the
line in miles, a is the attenuation constant per mile, and D is the
segment length in miles. If the line loss is 6 db or more the expression
is nearly equal to:

i?2 =

.46Z)a '

where a is the line loss in db per mile and D is the segment length
in miles. This assumes 4q:Z) is small compared to unity which is
usually the case.

The chance that the total unbalanced length will exceed about
l.XR is estimated at 1 per cent.

For far-end crosstalk (output-to-output) the same assumption as to
nominal segment length leads to the expression:

■Ji,-

The general expressions given for R^ suggest that a very long
segment might be permitted at some point in the line if the deviations
of the segments were properly restricted in other parts of the line.
The expressions given for far-end and near-end values of R^ were

i?2 = Si" -f Si'A,^ + 53^2^ + • • •.

If a very long segment at some point, such as a river crossing, were
permitted, this would increase the sum of the squares of the deviations
for some transposition section. For example S3 might be abnormally
large. R^ could be kept at some assigned value by limiting Si^, Si^, etc.
This procedure is not considered good practice because of the difficulty
of maintaining some parts of the line with very small deviations of
the segments from their nominal lengths.

OPEN-WIRE CROSSTALK 199

A very long segment has another effect on near-end crosstalk not
indicated by the above discussion. If there were no deviations in
any of the segments, the near-end crosstalk would be the vector sum
of a number of current elements of various magnitudes and phase
angles and the sum would be small due to a proper choice of these
magnitudes and angles in designing the transpositions. If a segment
deviates from its normal length, the magnitude of the crosstalk due to
the segment changes and the phase angle also changes. The phase
angles of the crosstalk values due to succeeding segments are also
changed since they must be propagated through the segment in
question. For ordinary deviations in segment lengths these effects
on the phase angles may be neglected.

Since transverse crosstalk is independent of transpositions occurring
in both circuits at the same point, it would appear from the above
discussion that the location of such transpositions need not be accurate.
This is not ordinarily a question of practical importance. If some
circuit combinations have both circuits transposed at a certain trans-
position pole there will usually be other combinations which have
relative transpositions at this pole. The transposition pole is of
importance, therefore, in connection with the latter combinations and
the same accuracy of location is required for all transposition poles.
A question of practical importance, however, is whether the above
rules for locating transposition poles properly limit the interaction
crosstalk. This is affected by transpositions in both circuits at the
same pole as well as by relative transpositions. In the following
discussion of this matter it is concluded that the effect of transposition
pole spacing irregularities on interaction crosstalk may be ignored at
frequencies now used for carrier operation.

The effect of deviations in segment length on interaction crosstalk
is indicated by Fig. 18. This figure indicates a short part of a parallel
between two long circuits a and b. A representative tertiary circuit
c is also shown. The transposition arrangements are like those of
Fig. 9B. In connection with the latter figure it was shown that the
interaction crosstalk would be very small if all segments had the same
length d. On Fig. 18, D is used to indicate the normal segment length
and the deviation of two segments from D is indicated by d. Since
the length A C equals the length CF, these deviations have no effect
on the transverse crosstalk which is controlled by the transposition at
C. The deviations affect the interaction crosstalk between the length
CF and length A C.

The circuit a has near-end crosstalk coupling with circuit c in the
length CF. This effect is normally practically suppressed by the

200

BELL SYSTEM TECHNICAL JOURNAL

transposition in a at E. Due to the deviation d of segment CE, the
near-end crosstalk between a and c in length CF will not be suppressed
but will be proportional to d. There will likewise be near-end crosstalk
between c and b in the length A C proportional to d. The two devia-
tions, therefore, introduce interaction crosstalk practically proportional
tod\

Fig. 18 — The eifect of deviations in segment length on interaction crosstalk.

Since there will be small deviations in numerous other segments of
circuit &, the deviation d in circuit a will introduce numerous other
interaction crosstalk paths similar to that discussed above. The
r.m.s. value of the total interaction crosstalk caused by deviations in
segment lengths may be roughly estimated as follows:

2FKyr\j^

AFKh^

^A6aD

<aD

where r is the r.m.s. deviation, L is the line length, D is the nominal
segment length and a is the line loss in db per mile, all distances being
expressed in miles. The above expression varies about as the 1.75
power of frequency and as the square of r. The corresponding

expression for transverse crosstalk, i.e., Fi^r -i/y^ varies as the first

power of frequency and of r. It follows that, if the rules for accuracy
of transposition pole spacing are relaxed or the maximum frequency is
raised, the effect of pole spacing on interaction crosstalk increases
more rapidly than the effect on the transverse crosstalk.

OP EN- WIRE CROSSTALK

201

For the range of frequencies and accuracy of pole spacing used in
practice, it has been found that the effect of pole spacing irregularity
on interaction crosstalk is not controlling. This is indicated by Fig. 19

Z 3000

A PAIRS 1 AND2,3AND4 90.53 MILES

CRAWFORDVILLE, ATLANTA
B PAIRS! AND 2, 3 AND 4 91 MILES

TERRE HAUTE, CHICAGO
C PAIRS 3 AND 4,23 AND 24 90.53 MILES
CRAWFORDVILLE, ATLANTA

/\-

v^P

--^^

0 5 10 15 20 25 30 35 40 45

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 19 — Far-end crosstalk caused by pole spacing irregularities.

which shows some measurements of output-to-output far-end crosstalk
between long circuits having transposition arrangements designed to
make the crosstalk due to type unbalance small compared to that due
to irregularities. The curves are about linear with frequency as would
be predicted if the effect of the pole spacing irregularities (and wire
spacing irregularities) on the interaction crosstalk is neglected. For
these particular curves, a knowledge of the pole spacing indicated that
pole spacing rather than wire spacing irregularities were controlling in
causing crosstalk.

The above discussion assumes that a transposition section is divided
by the transposition poles into segments all of the same nominal
length. It is sometimes economical to use segments of different
nominal lengths in the same transposition section. If the variation
among the segment lengths is consistent rather than accidental it may
be allowed for in the design of the transpositions.

In practice, segments of different lengths are used in the same
transposition section when it is desired to adapt for multi-channel
carrier frequency operation a few pairs on a line already having many
pairs transposed for voice-frequency operation. Such lines often have
existing transposition poles nominally spaced ten spans apart while
for the pairs retransposed for carrier operation it is necessary to space
the transposition poles about two spans apart. In such cases the cost

202 BELL SYSTEM TECHNICAL JOURNAL

of the carrier channels is appreciably increased if uniform spacing
between the new transposition poles is used. The transpositions in
the pairs retransposed for carrier operation must be coordinated with
the transpositions in the other circuits and it is necessary, therefore,
either to divide the ten spans into four approximately equal parts
with consequent expense in setting new poles at the quarter points or
to retranspose all the circuits on the line.

To avoid either of these expensive procedures, the new transposition
poles are nominally located in the manner indicated by Fig. 20. This

POLES
I 23456789 10 II

> o

o o c

1
1

) o o

1
1

o ^ o ^

1 1

I 1

II II

1 II 1

1 II II
1 II II

1 II II

1 1 ^ !

1 II 1

1 II II

1 II 1

1
1
1

1
1

-J

1
1

\
1

1
1

Fig. 20 — Location of extra transpositions in a ten-span segment of line.

figure shows ten-pole spans subdivided into four parts in order to
create three additional transposition poles. The figure indicates the
location of the new transposition poles and the possible methods of
transposing at these new poles. For some of the circuit combinations
the crosstalk within the ten-span interval is considerably greater than
if the four segments were equal in length. In each other ten-span
interval the crosstalk is likewise increased by a similar inequality in
segment length. Since all ten-span intervals are nominally alike,
considerable crosstalk reduction may be obtained by properly designed
transpositions located at the junctions of these intervals.

The use of segments of different lengths inherently decreases the
effectiveness of the transpositions in reducing crosstalk and adds to

OPEN-WIRE CROSSTALK 203

the complexity of the transposition design problem. Uniform seg-
ments are therefore used except in special circumstances.

Wire Spacing Irregularities

In the past there has been a tendency to permit wire spacing irregu-
larities in order to reduce the cost of construction and maintenance.
For example, "H fixture" crossarms formerly had special wire spacing
to permit the two poles to pass between pairs of wires and thus reduce
the length of the arms. Another example is that of resetting a pole
with a rotted base and reducing the spacing between crossarms to get
clearance between wires and ground. The development of repeatered
circuits and carrier current operation has increased the seriousness of
the crosstalk resulting from such irregularities and made such practices
generally undesirable.

There are, of course, unavoidable irregularities in wire spacing due
to variations in dimensions of crossarms, insulators and pole line
hardware and warping of crossarms. Corners and hills are other
causes since the crossarms at a corner and the poles on a hill are not
at right-angles to the direction of the wires. The most important
unavoidable spacing irregularity is, however, due to variations in wire
sag. Of recent years, limits have been set on wire sag deviations to
insure that this effect is properly limited during construction. The
main criterion adopted has been the difference in sag of the two wires
of a pair. This difference is a rough measure of the crosstalk increment
due to variations of the sag from normal. The crosstalk between
two pairs in a given span will be abnormal if the two pairs have different
sags even if there is no difference in sag for the two wires of a pair.
The crosstalk is usually more nearly normal, however, than in the
case of two pairs having the same average sag but different sags for
the two wires of a pair. As far as practicable, all pairs are sagged
alike in a given span.

The crosstalk between two pairs due to sag differences is computed
much like that due to pole spacing irregularities. The change in
crosstalk due to a known pole spacing deviation may, however, be
computed from the crosstalk coefficient while the change in crosstalk
due to a sag deviation is not related to the crosstalk coefficient in any
simple way. Two methods have been used to obtain constants for
calculation.

With the first method, crosstalk measurements were made on a
long line (about 100 miles) having small pole spacing and type un-
balance crosstalk. The r.m.s. of a number of crosstalk measurements
was determined for each particular type of pair combination, for

204 BELL SYSTEM TECHNICAL JOURNAL

example, for horizontally adjacent pairs. The r.m.s. of the sag
differences in a representative number of spans was also determined
for the two pairs of each type of combination. The two r.m.s. values
for any particular type of pair combination were called R and r. The
ratio of i? to r gave a constant k for estimating R from a known value
of r and for Lo, the particular length of line tested. For other line

lengths, R is estimated from the expression R = kryl — • Having

computed R, the chance of the crosstalk for any pair combination in a
long line lying in a given range may be estimated by probability
methods.

The second method of studying sag differences is more precise
although much more laborious. The change in crosstalk due to
introducing sag differences in but two spans is determined. The
poles are specially guyed to make it possible to adjust all the wires in
these spans to have practically the same sag. Turnbuckles are
installed at the ends of the two-span interval for this purpose. At
the center pole the wires were supported so as to slip readily and
equalize the sag in the two spans.

The phase and magnitude of the crosstalk is first measured for all
pair combinations with all wires at normal sag. The wires are termi-
nated in the same way as in the measurements of crosstalk coefficients.
From sag measurements on actual lines, a set of unequal sag values
for all the wires is then selected by probability methods and the
crosstalk remeasured. The vector difference between the values of
crosstalk before and after introducing unequal sags is then determined.
This process is repeated a large number of times in order to cover the
range of sag conditions encountered in practice. An r.m.s. value of
the change in crosstalk due to sag difference is then determined for
each pair combination and related to the r.m.s. sag difference per pair.
This permits the probable crosstalk in a long line to be estimated and
the importance of sag difference crosstalk to be determined. The
two methods of study were found to be in general agreement. The
second method has been extensively used to study proposed new wire
configurations.

Drop Bracket Transpositions

An ideal transposition would cross the two sides of a circuit in an
infinitesimally small distance, there being no displacement of the
wires from their normal positions on either side of the transposition.
The point-type transposition indicated by Fig. 21 is close enough to
the ideal for practical purposes. Its deviation from the ideal requires
little consideration in transposition design. To avoid cutting the

OP EN -WIRE CROSSTALK

205

wires, one wire is raised about 3/4 inch and the other lowered this
amount at the transposition point. The drop bracket transposition

Fig. 21 — Point-type transposition.

illustrated by Fig. 22 is considerably cheaper but the displacement of
the wires is much greater. The effect of this displacement is important
and must be especially considered in transposition design.

If all the spans adjacent to a drop bracket were of the same length

O©

Fig. 22 — Drop-bracket transposition.

206 BELL SYSTEM TECHNICAL JOURNAL

and all wires could be kept under the same tension, the effect of drop
brackets on crosstalk would be consistent and could, theoretically, be
made negligible by a suitable transposition design.

There is, however, an accidental crosstalk effect. This effect is
partly due to the fact that it is more difficult to avoid deviations from
normal sag in the spans adjacent to drop brackets than in normal spans.
The main effect, however, is thought to be due to inequalities in the
lengths of the spans adjacent to drop brackets.

The crosstalk in such a span is very nearly proportional to the
length of the span times a constant or "equivalent crosstalk coeffi-
cient." The usual crosstalk coefficient can not be used because the
wires are not parallel.

Fig. 23-A indicates two long circuits, one circuit being transposed
on drop brackets at the first and third quarter points of the short
length D. The lengths of the spans adjacent to the drop bracket
transpositions are indicated by di to d^. The equivalent far-end
crosstalk coefficient for the span preceding a transposition bracket is
Fi and that for the span following the bracket is F2. (Fi and F2 are
usually quite different.) The total far-end crosstalk (output-to-
output) due to the four spans is (very nearly) :

K{Fidi - F^di - Fidi + Fidi),

where K is the frequency in kilocycles.

If the four spans were equal the crosstalk would be zero (very
nearly). The actual value of the crosstalk is a matter of chance
since the deviations of the four spans from the normal length are a
matter of chance. These deviations cause a chance increase in the
near-end crosstalk as well as in the far-end crosstalk.

This effect has been studied experimentally by using transposition
designs which suppressed the consistent effect. The pole spacing
effect was minimized by using very accurate spacing. The wire sag
effect was allowed for by comparing similar pair combinations trans-
posed alike except that dead-ended point transpositions were compared
with drop bracket transposition. Due to the great number of trans-
positions necessary at carrier frequencies it was found that the acci-
dental drop bracket effect was important at these frequencies. In
recent years, point-type transpositions have been extensively used on
lines transposed for long-haul carrier systems.

When, for economic reasons, a transposition system is designed for
use with drop bracket transpositions, the consistent crosstalk effect
must be considered in the transposition design. The equivalent
crosstalk per mile for a span adjacent to a drop bracket must be

OP EN -WIRE CROSSTALK

207

determined for each pair combination. Approximate methods of
computation have been worked out for doing this and checked against
measurements. The computations are involved in connection with
far-end crosstalk since the "tertiary effect" is controlling. Since the

1^ A\

V VI

) V

d| I d2 I

(A)

I d3 I d4 I I

(B)
Fig. 23 — Effect of drop brackets on crosstalk.

summation of crosstalk due to drop brackets is a consistent effect,
"drop bracket type unbalances" can be worked out and used in
transposition design. This matter is so complicated, however, that
the practical method of design is to first practically ignore the drop
bracket efTect and then check the design to determine whether this
effect has been properly suppressed.

Certain rules are adopted, however, to ensure that the transposition
arrangements are properly chosen to avoid the larger drop bracket
effects. Fig. 23-B indicates an arrangement of transpositions for two

208

BELL SYSTEM TECHNICAL JOURNAL

pairs in a short length of Hne which, with point transpositions, would
have very low crosstalk. At points B and E both circuits are trans-
posed alike. With point transpositions the near-end crosstalk in the
two spans adjacent to one of these pairs of transpositions would be
NK2d, where d is the span length, N the near-end crosstalk coefhcient
and K the frequency in kilocycles. For drop bracket transposition
the crosstalk would be K{Ni + iVz)^ or a change of K{Ni -f A^2
- 2N)d.

The transpositions are so arranged that the crosstalk in the two
spans at B tends to add to that in the two spans at E. With drop
brackets at B and E the major crosstalk in this length of line would
be twice the above change since the crosstalk with point transpositions
is very small.

ALSO TRANSPOSITIONS IN BOTH
PAIRS AT THIS POINT AND
/"each Va f^lLE THEREAFTER

J^MILE

/ y

n\\

//\

0 5 10 15 20 25 30 35 40 45 50

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 24 — Near-end crosstalk with and without drop brackets.

If the arrangement of Fig. 23-B is reiterated in a long line, the total
increase in the crosstalk due to drop brackets at such points as B and
E may be marked. It may be noted that the crosstalk in the two
spans at A tends to cancel the crosstalk in the two spans at C and
likewise there is cancellation at D and F. Drop brackets may,
therefore, be used at points ^, C, D and F without a consistent increase
in crosstalk. Arrangements like those at B and E of Fig. 23B should
be avoided in transposition design involving drop brackets.

The change in the crosstalk due to drop brackets is not necessarily
an increase. Fig. 24 shows an arrangement of transpositions in an
eight-mile line and three crosstalk frequency curves. Curve A shows

OPEN-WIRE CROSSTALK

209

the calculated near-end crosstalk for ideal point transpositions.
Curves B and C show the calculated and observed near-end crosstalk
for drop bracket transpositions. The curves show that the drop
bracket effect can be calculated quite accurately and that it may
reduce the total crosstalk. In the general case, it is impractical to
take much advantage of this reduction effect because a marked re-
duction for one combination of circuits is likely to result in an increase
for some other combination and because a reduction of crosstalk in
one part of the line may increase the vector sum of crosstalk elements
from all parts of the line.

Wire Configurations

The crosstalk coefficients for the various pair combinations may be
altered by changing the configuration of the wires. Therefore, the
crosstalk for a given transposition design and a given accuracy of
transposition pole spacing irregularity may also be altered. The
crosstalk due to sag differences also depends on the wire configuration.
It is important, therefore, to choose a configuration most desirable
from the crosstalk standpoint. Such an optimum configuration
requires the fewest transpositions and least accuracy of pole spacing
for a given maximum frequency and given permissible values of
crosstalk coupling.

Various "non-inductive" arrangements of wire configurations have
been suggested and tested. Such arrangements may appear to have
possibilities but their study to date has indicated that they are im-
practicable for more than a few pairs on a line.

O 0

3 4

o o

3 4

O O

1 3

O O

2 4

Fig. 25 — "Non-inductive" arrangements for two pairs of wires.

Fig. 25 illustrates several suggested arrangements for two pairs.
Arrangement A is often called a square phantom. If pair 1-2 is the
disturber and there are equal and opposite currents in wires 1 and 2
there will be no voltages induced in either wire 3 or wire 4 because
either of these wires is equally distant from wires 1 and 2. Since
wires 1 and 2 are not equally distant from the ground, the currents

210 BELL SYSTEM TECHNICAL JOURNAL

in these wires may be not quite equal and opposite. As a result,
voltages will be induced in wires 3 and 4 but these will be equal and
there will be no crosstalk current in pair 3-4. By the reciprocal
theorem the crosstalk between the two pairs will also be zero when
pair 3-4 is the disturber.

Arrangement B is nearly non-inductive. In this case if pair 1-2 is
the disturber and the currents in the two wires are not quite equal and
opposite due to the presence of the ground, unequal voltages will be
induced in wires 3 and 4 and there will be a crosstalk current in this
pair. This effect could be minimized by transposing both pairs at
the same points. They would not require relative transpositions since
equal and opposite currents in pair 1-2 will induce no voltage in either
wire 3 or wire 4.

With pair 3-4 as the disturber, equal and opposite currents will
result in equal voltages induced in wires 1 and 2. These voltages
cause a phantom current in phantom 1-2/3-4. This phantom current
will divide between wires 3 and 4 but can not induce unequal voltages
in wires 1 and 2 because 1 and 2 are equally distant from either 3 or 4.
The crosstalk coefficient is, therefore, zero both for the direct effect
and for the indirect effect of the phantom. However, the indirect
effect of the ground or other conductors is not zero and may require
transpositions.

Arrangement C is non-inductive for direct crosstalk. It is not
non-inductive in regard to the indirect effect of the phantom 1-2/3-4.
Equal and opposite currents in pair 1-2 induce equal voltages in wires
3 and 4. The resulting equal phantom currents in wires 1 and 2 of
phantom 1-2/3-4 will induce unequal voltages in pair 3-4.

When there are many pairs on a line it is not possible to make all
combinations strictly non-inductive even for direct crosstalk. With
perfect wire spacing the larger values of direct crosstalk per mile
could be greatly reduced, however, and appreciable reductions could
be obtained in the indirect effect which is usually controlling in far-end
crosstalk.

Wire sag deviations must be considered, however. If a given
number of "non-inductive" pairs are placed in the pole head area
normally occupied by the same number of pairs with conventional
configuration, the crosstalk due to sag deviations is likely to be more
serious with the "non-inductive" pairs than with conventional pairs.
For the same pole head area, the number of transpositions and,
therefore, the "pole spacing" crosstalk could be reduced if non-
inductive arrangements were used. The tests to date indicate,
however, that the total crosstalk would not be reduced because of
increased "sag difference" crosstalk.

OPEN-WIRE CROSSTALK

211

The mechanical problem of supporting the wires of the "non-
inductive" arrangements is considerable if serious increases in crossarm
and hardware costs are not to be incurred. This objection seems at
present to override the possible advantages of (1) fewer transpositions
for a given pole head area and crosstalk result, or (2) fewer trans-
positions and lower crosstalk with a greater pole head area.

Another possibility is the use of non-parallel wires. It is possible to
arrange two pairs of wires in such a way that they have a certain direct
crosstalk per mile at one end of a span and the value at the other end of
the span is about equal and opposite. The net direct crosstalk per mile
integrated over the span is zero or small. An example of this is the
barreled square formerly used abroad. Fig. 26 illustrates this arrange-

CROSS

SECTIONS OF

WIRES

POLE 1

9
o

o
10

o
1

MID-SPAN

2
o

o
9

o
10

POLE 2

o
9

2
o

10
o

Fig. 26 — Two pairs of wires in different barreled squares.

ment. The wires are arranged in groups of four, each four being
arranged on the corners of a square. The two wires of a pair are on
diagonally opposite corners of a square. Each pair is given a quarter
turn in each span. For simplicity only two pairs in different four-wire

212 BELL SYSTEM TECHNICAL JOURNAL

groups and one span are shown. The two pairs shown are nearly
"non-inductive" for direct crosstalk in this span.

Consideration has been given to applying this principle to a number
of pairs in order to reduce the crosstalk coefficients. Since all the
crosstalk coefficients could not be made very small, transpositions
would be needed. The experience to date indicates that this method
does not look attractive because it is not very effective in reducing the
indirect crosstalk, the mechanics of transposing are difficult, the
variations in sag are likely to be abnormal and the system is compli-
cated.

There remains the simple method of improving the configuration
of the wires in a given pole head area by reducing the spacing between
the wires of a pair and increasing the spacing between wires of different
pairs.

The crosstalk per mile between pairs is evidently reduced by this
procedure since the two wires of a pair are approaching the ideal of
being equally distant from every other conductor. The "sag difference
crosstalk" is also reduced and higher frequencies may be used for a
given crosstalk result. Fig. 27-A and Fig. 27-B indicate a 20-wire
line with the wire spacing used in the past and also the configuration
commonly used today on lines where heavy carrier development is
involved. The spacing between the two wires of a pair has been
reduced from 12 inches to 8 inches and the spacing between pairs
correspondingly increased.

It was not possible to reduce the spacing of the pole pairs and for
this reason they are unsuited for the higher carrier frequencies and it
is sometimes uneconomic to string them. For such cases the crossarm
indicated by Fig. 27-C may be considered. The 8-inch spacing of
pairs is retained but the distance between pairs is further increased.
With this last crossarm, phantom circuits are not superposed on the
8-inch pairs since their use results in greater crosstalk between the
pairs and restricts the possibilities of multi-channel carrier operation.

The crossarm with 8-inch pairs and pole pairs may be used on lines
where multi-channel carrier operation is not employed. In such
cases, the 8-inch pairs may be phantomed. Since the average spacing
between the side circuits of such a phantom is not reduced by the
8-inch spacing, the crosstalk between the phantom circuits is about
the same as with the 12-inch pairs. The crosstalk from a side circuit
into a phantom is somewhat reduced because of the reduced spacing
of the pairs. For a given pole head area it does not appear practicable
to devise a configuration which will result in marked reductions in the
susceptibility of both phantoms and side circuits to crosstalk and

OPEN-WIRE CROSSTALK

213

5^'Wl2'^— j— 12'^— 1-«— l2"-^9|-"-i4- \8j »|-9|'U^I2"-*|-— 12"— 1-<— l2'U|5g

Q" ^' ^'

Fig. 27 — Configurations of open-wire lines.

214

BELL SYSTEM TECHNICAL JOURNAL

noise The "square phantom" indicated by A of Fig. 25 has theo-
retical possibiUties but studies of the effect of wire spacing deviations
make this arrangement appear impracticable.

The proposal to reduce the spacing of the wires of a pair from the
historic value of 12 inches naturally raised the question of swinging
contacts. However, extensive experience with 8-inch spacing has
shown no appreciable increase in the number of wire contacts. This
applies to lines where ordinarily the span length did not exceed about
150 feet. With long span crossings, crossarms were supported from
steel strand at intervals of 260 feet or less.

The effectiveness of the reduction in wire spacing is indicated by
the following table. The table shows the measured near-end and
far-end crosstalk coefficients for important circuit combinations and
for the two-pole head diagrams of Figs. 27-A and 27-B.

Crosstalk Per Mile Per Kilocycle — 104-Mil Conductors

Pair Combination

Near-End Crosstalk

Far-End Crosstalk

12-Inch

8-Incli

12-Inch

8-Inch

1-2 to 3-4

3-4 to 7-8

974
133

653
40

549

163
55

107

439

47
326
18
288
78
28
55

77
66
58
155
35
43
75

1-2 to 11-12

1-2 to 13-14

30
24

3-4 to 13-14 ....

1-2 to 21-22

1-2 to 23-24

3-4 to 23-24

General Transposition Design Methods
The preceding discussion will indicate that transposition design
involves much more than consideration of the locations of the trans-
positions.

In practical design, the first step is to estimate the crosstalk due to
unavoidable pole spacing and wire spacing irregularities for the
configuration of wires under consideration and for a wide frequency
range. This crosstalk represents the best that can be done with an
ideal transposition design. It must be kept in mind that great
precision is impracticable. The pole spacing of a line may change
from time to time due to minor reroutings caused by highway changes,
etc. The wire sag differences change with temperature and are
affected by sleet.

If two long circuits are on adjacent or nearby pairs in one repeater
section, they should, as far as practicable, be routed over non-adjacent

OP EN- WIRE CROSSTALK 215

pairs in other repeater sections in order to minimize the overall cross-
talk between these two circuits. This crosstalk will usually be largely
due to those parts of the parallel where the circuits are on adjacent
or nearby pairs, since the pole line seldom has enough pairs to make it
practicable to keep any two circuits far apart for a large proportion
of the total parallel. It is important, therefore, to strive for the lowest
possible crosstalk between adjacent or nearby pairs even though this
requires permitting higher crosstalk between widely separated pairs
than would otherwise be necessary.

For the adjacent or nearby pairs with naturally high crosstalk,
limits on the type unbalance crosstalk are set which make this type of
crosstalk small compared with that due to irregularities. Since the
type unbalance crosstalk varies with frequency and, in general,
increases with frequency, these limits are imposed only for the range
of frequency which the line will be required to transmit. It is not
advisable to go beyond this, since more severe limits require closer
spacing of transpositions and the increased number of transpositions
would make the "pole spacing" irregularity crosstalk larger. For
the well-separated pairs with naturally lower crosstalk, the type
unbalance crosstalk rather than the irregularity crosstalk may be
allowed to control with the same idea in mind of requiring a minimum
number of transposition points.

Fig. 28 indicates the method used generally in the Bell System for
arranging transpositions with 32 transposition poles. The arrange-
ments shown are called fundamental types. They are iterative, i.e.,
if the first two-interval length is transposed at the center, each fol-
lowing two-interval length is likewise transposed, etc. Various other
arrangements called hybrid types are possible but in the long run
there appears to be no advantage from their use except in the case of
side circuits of phantoms. In this case the transposition pattern may
change when the side circuit changes pin positions at a phantom
transposition.

The fundamental types may be extended to involve 64, 128, 256,
etc., transposition poles. Types involving 128 transposition poles are
often used.

A long line, say 100 miles, is divided into short lengths called
transposition sections. With the latest transposition designs, sections
having 128, 64, 32, 16 and 8 transposition poles are provided. The
nominal lengths of these sections vary from 6.4 to .25 mile. The
purpose of these sections is to provide an approximate balance against
crosstalk (and induction from power circuits) in short lengths and
thus to allow for unavoidable discontinuities in the exposure between

216

BELL SYSTEM TECHNICAL JOURNAL

circuits such, for example, as points where circuits branch off the line.
Transposition arrangements must be chosen for each circuit in each
type of section to ensure this approximate crosstalk balance.

Fig. 28 — Fundamental types for 32 transposition poles.

Certain lines have few, if any, discontinuities and a succession of the
longest type of section is used. To improve the effectiveness of the
transpositions, junction transpositions are used at the junctions of
successive similar sections. For such lines it would be more effective
to use longer transposition sections and not require that all circuits be
approximately balanced in a short length. Such a special design
would be impracticable, however, since it would be too inflexible in
regard to circuit changes, etc.

In choosing the transposition arrangements for a section it must
be kept in mind that the object is to meet certain crosstalk limits for
a succession of sections considering both type unbalance and irregu-
larity crosstalk. The method of procedure is discussed below.

OPEN-WIRE CROSSTALK 217

Evolution of Transposition Designs

In designing transposition systems it must be kept in mind that
much of the crosstalk is due to irregularities and is a matter of chance.
Theoretically the crosstalk elements due to all of the various irregu-
larities might chance to add directly. This is highly improbable and
if the design were based on making this limiting condition satisfactory,
the expense would be very great. Practically, therefore, the designs
are based on exceeding a tolerable value a small percentage of the
time. If, in practice, the tolerable value happens to be exceeded and
this is not found to be due to an error in construction, the unfortunate
adding up of crosstalk elements can be broken up by a different
connection of circuits at the offices.

The tolerable values commonly chosen are 1000 crosstalk units
(60 db) for open-wire carrier circuits and 1500 units (56 db) for voice-
frequency open-wire circuits, which tend to have more line noise than
cable or carrier circuits. These limits apply to the crosstalk between
terminating test boards with the circuits worked at net losses of about
9db.

Before proceeding with the design of the individual transposition
sections which are but a few miles long, it is evidently necessary to
determine what part of the overall limit can properly be assigned to an
individual section. Assumptions must first be made as to typical
and limiting lengths in which circuits are on the same pole line and
in which adjacent or nearby circuits continue in this relation. A
representative repeater layout must then be chosen. The repeater
layout is very important, since the crosstalk in each repeater section
is propagated to the circuit terminal and amplified or attenuated,
depending on the arrangement of the repeaters. As a matter of fact,
the layout of repeaters must be governed to a considerable extent by
crosstalk considerations.

On the assumption that the relative magnitudes and phase relations
of the crosstalk couplings in the various repeater sections are a matter
of chance the tolerable crosstalk in a single repeater section can be
estimated by the use of probability laws. Similarly the tolerable
value for any part of the repeater section can be estimated. These
probability methods apply very well to crosstalk due to irregularities.
Type unbalance crosstalk is systematic, however, and in assigning
tolerable values of type unbalance crosstalk in a transposition section,
it is necessary to consider how the crosstalk values for various trans-
position sections may add up.

It is not likely that there will be systematic building up of type
unbalance crosstalk in successive repeater sections and, therefore, the

218 BELL SYSTEM TECHNICAL JOURNAL

tolerable crosstalk per repeater section may be estimated by probability
methods. The total of the irregularity crosstalk and the type un-
balance crosstalk in a repeater section is a matter of chance and may
be estimated from probability theory. Conversely, the part of the
tolerable crosstalk which may be assigned to type unbalance crosstalk
may be estimated. As noted above, the allowance for type unbalance
crosstalk for adjacent or nearby pairs is usually made so small that
irregularity crosstalk controls the total. The maximum permissible
carrier frequency is, then, the frequency at which the irregularity
crosstalk just reaches the tolerable value. Having determined toler-
able values of type unbalance crosstalk for a repeater section for the
various pair combinations, tolerable values for the individual trans-
position sections must be determined.

If a repeater section involves a number of different types of trans-
position sections it is not likely that there will be a systematic building
up of type unbalance crosstalk. Factors are, therefore, worked out
to relate the crosstalk in a succession of similar transposition sections
to that in one section. Numerous factors are required since they
depend upon the transpositions at the junctions of the sections. A
study of such factors indicates values which it is reasonable to assign
to an individual transposition section in order to avoid excessive type
unbalance crosstalk in a complete repeater section.

In the case of a voice-frequency transposition system, both near-end
and far-end type unbalance limits must be set. The far-end limits
are usually easily met. In the case of a transposition system for
carrier systems, far-end crosstalk is controlling and the far-end type
unbalance limits are important. The "reflection crosstalk" previously
discussed depends, however, on both the magnitude of the near-end
crosstalk and on the impedance mismatches. Information on the
degree to which it is practicable to reduce these mismatches must be
available in order to set limits on near-end type unbalances at carrier
frequencies.

Pairs used for carrier systems are usually also used for voice-
frequency telephone systems and in designing transpositions for these
pairs crosstalk limits suitable for both types of systems must be met.
In practice, an existing line may have only a part of the pairs retrans-
posed for carrier operation and in designing a system of transpositions
for such retransposed pairs limits must be set for the crosstalk at
voice frequencies between the retransposed pairs and the pairs not
retransposed.

It has been the practice to transmit certain carrier telegraph fre-
quencies in the opposite directions used for these frequencies in

OPEN-WIRE CROSSTALK 219

connection with carrier telephone, or, in some cases, program trans-
mission circuits. At these frequencies near-end crosstalk limits must
be set so as to limit the induced noise from the carrier telegraph.

When the type unbalance crosstalk limits are finally determined,
the transposition designer must attempt to meet the requirements for
all circuit combinations and all the transposition sections. It may be
that the requirements can not be met and consideration must be given
to modifications in the nature of the transmission systems. A vast
amount of such preliminary transposition design work has been
necessary in order to evolve the present transposition systems and
transmission systems.

Such studies led to the development of non-phantomed circuits
with 8-inch spacing since they indicated that multi-channel long-haul
carrier operation on all pairs on a line was, in general, impracticable
from the crosstalk standpoint with 12-inch phantomed pairs.

It may be noted that there are also difficulties in the crosstalk
problem when 12-inch phantomed pairs are used for voice-frequency
repeatered circuits. These circuits have a crosstalk advantage over
carrier circuits in that the frequency is lower but they have an off-
setting disadvantage in that they use the same frequency range in
both directions. This makes the near-end crosstalk directly audible
to the subscriber. As previously discussed the near-end crosstalk is
inherently greater than the far-end crosstalk and, for this reason,
practicable designs of multi-channel carrier systems do not allow near-
end crosstalk to pass to the subscriber, the path being blocked by
one-way amplifiers. While it takes fewer transpositions to control
the type unbalance effects with voice-frequency transposition designs,
for a given length of parallel the difficulties with crosstalk due to
irregularities are about as great as with designs for multi-channel
carrier operation.

The simple example of Fig. 29 illustrates the reasons for the diffi-
culties with near-end crosstalk with the voice-frequency designs for
12-inch spaced pairs. It also illustrates the method of deducing the
permissible crosstalk per repeater section as discussed above.

This figure indicates two paralleling repeatered circuits, each having
six repeater sections of 10 db loss and five repeaters of 10 db gain.
The net loss of each circuit is, therefore, 10 db. The near-end crosstalk
values in the six sections are indicated by Wi to «6- The crosstalk
coupling at A due to m is just equal to W2 since there is no net loss or
gain in either circuit between A and B. There is also no net loss or
gain between A and C, A and D, A and E or A and F. The total
crosstalk coupling at A is, therefore, the vector sum of the six values

220 BELL SYSTEM TECHNICAL JOURNAL

Hi to We- If the crosstalk is due to irregularities the exact values of
11 can not be calculated but from the data collected on the crosstalk
due to irregularities, the r.m.s. of all possible values may be estimated,

W 10 DECIBELS NET LOSS H

KlOdbLOSS*- ■• — lOdb — »■ -• — lOdb — *■ •• — lOdb — •■ ■• — lOdb-

E F

Fig. 29 — Crosstalk between repeatered circuits.

Letting n equal the r.m.s. value of Wi, etc., and using probability
theory we may write:

i?2 = ^2 + ^2^ + • • • + re^,

where R is the r.m.s. of all possible values of the near-end crosstalk
at ^. If ri = r2, etc.

R = rV6.

The chance of the overall crosstalk deviating from R by any specified
amount may be estimated by probability methods. It will be noted
that the crosstalk in six repeater sections tends to be more severe
than that in one section by \^ or, in other words, that the crosstalk
varies as the square root of the length. If the use of repeaters were
avoided by using more copper, for the same overall loss the crosstalk
would be practically the same as with the repeatered circuits. With
the arrangement of repeaters shown it is not the use of repeaters which
causes the increase in crosstalk but rather the increase in circuit length
without corresponding increase in circuit loss. For a given circuit
length, circuit loss and wire size, other arrangements of repeaters may
cause greater or less crosstalk.

If the repeaters of Fig. 29 are spaced farther apart, say 15 db
instead of 10, there will be three line repeaters of 15 db gain each and
terminal repeaters will be necessary to supply a terminal gain of 5 db
in order to obtain a net loss of 10 db. The near-end crosstalk would
be reduced by about V4 -t- \'6 or 1.8 db because there are only four
repeater sections but the terminal repeaters would ampUfy the near-end
crosstalk by 5 db. The net increase would be 3.2 db. From the
standpoint of near-end crosstalk, it is thus seen that close spacing
between repeaters is very desirable.

OPEN-WIRE CROSSTALK 221

In Fig. 29 the output-to-output far-end crosstalk in each repeater
section is indicated by /i to /e- The transmission path through any
one of these crosstalk couplings is (for like circuits) a loss 10 db greater
than the value of the coupling expressed as a db loss. With the
repeater arrangement of the figure, the far-end crosstalk paths are
attenuated by 10 db while the near-end crosstalk paths are not
attenuated. Furthermore, the far-end crosstalk paths ordinarily
introduce greater losses than the near-end paths. With greater
spacing between repeaters, the near-end crosstalk is amplified but the
far-end crosstalk (for like circuits) is still attenuated by the net loss
of the circuits. At a given frequency the near-end crosstalk between
such "two-wire" circuits is, therefore, much greater than the far-end
crosstalk.

Review

Evidently the problem of keeping crosstalk between open-wire
circuits within tolerable bounds is by no means a simple one. As we
have seen, the work begins with consideration of complete circuits
(telephone, program transmission or carrier telegraph) which may be
hundreds or even thousands of miles long. The total crosstalk
allowance for such long circuits must first be broken down into allow-
ances for the various sections of line between repeaters and then into
allowances for the individual transposition section, these individual
sections ranging from less than 1/4 to about 6 miles in length.

Then bearing in mind that irregularities in pole spacing and in wire
configuration set limits to crosstalk reduction which it is not practicable
to overcome by transpositions, the crosstalk designer determines by
computation whether, when considering these irregularity effects
alone, the crosstalk requirements for the individual transposition
sections can be met. If these requirements can not be met he must
either have the general circuit layout altered so that, for example,
the repeater gains will be more favorably disposed from the standpoint
of crosstalk, or he must alter the pole head configuration so that the
electrical separation between the circuits will be increased.

Having obtained an overall circuit layout and a configuration of the
wires which makes it possible to attain the desired overall crosstalk
results, the design of the transpositions proper is undertaken. In
this work the transposition designer makes every effort to keep the
number of transpositions at a minimum. He does this partly to save
money but more particularly because he recognizes that more than
enough transpositions do harm rather than good by increasing the
number of pole spacing irregularities.

222 BELL SYSTEM TECHNICAL JOURNAL

In dealing with the problems of crosstalk coupling between open-
wire circuits, consideration must be given not only to the direct effect
of one circuit on another but also to the indirect effect of the other
circuits on the line. What happens is that the disturbing circuit
crosstalks not only directly into the circuit under consideration but
also into the group of other circuits and thence into the disturbed
circuit. The name "tertiary" circuit has been given to this group of
circuits although it is not in reality one circuit but rather any or all
of the possible circuits which may be formed of the different wires.
The system of transpositions must, therefore, not only substantially
balance out the direct couplings between disturbing and disturbed
circuits but must also substantially balance the couplings from the
disturbing circuit into the "tertiary" circuit and from this "tertiary"
circuit into the disturbed circuit.

Reflections of the electrical waves also add interest and complexity
to the problem. Such reflections tend to increase crosstalk because
the electrical waves which are changed in direction as a result of
reflections crosstalk differently, and in many cases more severely,
into neighboring circuits than do the waves traveling in the normal
direction. The most important reflections occur at junctions between
lines and office apparatus. The possibility of other reflections must
also be considered, however, at intermediate points in the line which
might be caused by inserted lengths of cable, change in spacing of
wires, etc.

In working out the transposition designs, the fact that crosstalk
between two paralleling circuits tends to manifest itself at both ends
is of great importance. At the "near end" crosstalk coming from
the disturbed circuit in a direction opposite to the transmission in the
disturbing circuit must be considered. At the "far end" crosstalk
coming in the same direction as the transmission in the disturbing
circuit must be considered.

For telephone circuits which use the same path for transmission in
both directions, the "near-end" crosstalk is considerably more severe
than the "far-end" for two reasons: (1) The crosstalk per unit length
of the paralleling circuits is greater; (2) the gains of the repeaters
especially augment the "near-end" crosstalk. Voice-frequency open-
wire telephone circuits have always been worked on this "one-path"
basis and are good examples of circuits in which "near-end" crosstalk
is controlling and must be given principal consideration in working
out transposition designs.

In the case of carrier circuits, it was found early in the development
that if these circuits were worked on a one-path basis, the crosstalk

OPEN-WIRE CROSSTALK 223

would be prohibitively great. Consequently, carrier circuits are now
designed to operate on a two-path basis. Two separate bands of
frequencies are set aside, each being restricted, by means of one-way
ampUfiers and electrical filters, to transmission in one direction only.
Each telephone circuit is then made up of two oppositely directed
channels, one in each frequency band. Thus, direct "near-end"
crosstalk is kept from passing to the telephone subscribers. Conse-
quently, the "near-end" type of crosstalk needs to be considered
only with respect to that portion which arises from electrical waves
reflected at discontinuities in the circuits, which effects have already
been mentioned.

In practice a pole line may have some of the pairs very frequently
transposed to make them suitable for carrier frequency operation and
other pairs less frequently transposed and suitable only for voice-
frequency operation. A system of transpositions must permit any
arrangement of the two types of pairs which may be found economical
for a given line and layout of circuits. Each pair must meet limits
for near-end and far-end crosstalk to any other pair which may
crosstalk into it in its frequency range. Pairs used only for voice
frequencies are usually phantomed and transpositions must, of course,
be designed for the phantom circuits as well as the side circuits. The
design of a transposition system is, therefore, extraordinarily compli-
cated and tedious and, to paraphrase the Gilbert and Sullivan police-
man, "A transposer's lot is not a happy one."

Bibliography

The published material on the matter of open-wire crosstalk and
transposition design appears to be very limited. The following papers
are of interest:

The Design of Transpositions for Parallel Power and Telephone Circuits, H. S.

Osborne. Trans, of A. I. E. E., Vol. XXXVII, Part II, 1918.
Telephone Circuits with Zero Mutual Induction, Wm. W. Crawford, Trans, of A. I.

E. E., Vol. XXXVIII, Part I, 1919.
Measurement of Direct Capacities, G. A. Campbell. Bell System Technical Journal,

July, 1922.
Propagation of Periodic Currents Over a System of Parallel Wires, John R. Carson

and Ray S. Hoyt. Bell System Technical Journal, July, 1927.
On Crosstalk Between Telephone Lines, M. Vos. L. M. Ericsson Review, English

Edition, 1930, Vol. 7.
Application of High F"requencies to Telephone Lines, M. K. KupfmuUer. Presented

Before International Electrical Congress, July, 1932.
Probability Theory and Telephone Transmission Engineering, Ray S. Hoyt. Bell

System Technical Journal, Jan., 1933.

224 BELL SYSTEM TECHNICAL JOURNAL

APPENDIX

Calculation of Crosstalk Coefficients

This appendix will first cover methods of calculating the coefficients
of transverse crosstalk coupling. It is necessary to calculate both
near-end and far-end crosstalk coefficients which involve both direct
and indirect components of transverse crosstalk coupling. Coefficients
for the direct and for the indirect components will be derived separately
and then combined to obtain the total coefficients.

Ordinarily, the indirect effect cannot be readily computed with good
accuracy and the total coefficients are usually measured. As previ-
ously noted, the method of computing the indirect effect can be used
with fair accuracy, however, and it is useful in cases where measure-
ments are impracticable.

The crosstalk between frequently transposed circuits may be
calculated with the aid of the above coefficients of transverse coupling
and in addition an "interaction crosstalk coefficient" relating to
interaction crosstalk coupling of the most important type. The
relation of this interaction coefficient to the far-end coefficient of
transverse coupling is also discussed herein.

Direct Crosstalk Coefficients

Figure 30 indicates the definitions of the direct crosstalk coefficients

used in computing the direct component of the transverse crosstalk

coupling. This figure shows a thin transverse slice in a parallel

between two long circuits a and h, the thickness of the slice being the

■*

Zcx — ^

TO LONG

CIRCUITS

TO LONG
CIRCUITS

— Zb

J^r\

■« '

Zb —

-*1

P'ig. 30 — Crosstalk in a single infinitesimal length.

infinitesimal length dx. Circuit a is energized from the left, the
current entering dx being la. Propagation of la through dx results in
near-end and far-end currents in and i/ in circuit b at the ends of dx.
Since the coefficients are the crosstalk per mile per kilocycle, the
near-end coefficient A^ and the far-end coefficient F may be expressed

OPEN-WIRE CROSSTALK

225

as follows:

N = limit of

F = limit of

la ' Kdx
if 10«

la Kdx

as dx approaches zero.

where K is the frequency in kilocycles. For circuits of different
characteristic impedances Za and Zj, the above current ratios should
be multiplied by the square root of the ratio of the real parts of Zb
and Za. This correction is not included in the expressions for A^ and
F derived below.

Figure 31 indicates the equivalent electromotive forces which, if
impressed on the disturbed circuit h, would cause the same direct
crosstalk currents as the electric and magnetic fields of the disturbing
circuit. The series and shunt electromotive forces Vm and Ve corre-

n) )^e Zb— *

Fig. 31 — Equivalent e.m.f.'s in a disturbed circuit.

spond to the magnetic and electric components of the field and cause
crosstalk current im and ie. These currents are about equal in magni-
tude and they add almost directly at the near end of the length dx
and subtract almost directly at the far end. The near-end coefficient
is, therefore, inherently much greater than the far-end coefficient.

To calculate i^ the crosstalk current due to the electric field of
circuit a, it is necessary to know the shunt voltage Ve- This depends
on the charges on the wires of circuit a in the length dx. These
charges are due to a voltage V impressed on the left-hand end of
circuit a which may be remote from the length dx. Since it is desired
to transmit on the metallic circuit a and not on the circuit composed
of its wires with ground return, care is taken to "balance" the im-
pressed voltage, i.e., this sending circuit has equal and opposite
voltages between its two sides and ground with circuit a disconnected.

226 BELL SYSTEM TECHNICAL JOURNAL

The impressed voltage V is propagated to the left-hand end of dx.
Letting Va be the voltage across circuit a at this point, it will be
shown that Va would be balanced except for the effect of interaction
crosstalk which is excluded from consideration for the present. Desig-
nating the wires of circuit a as 1 and 2, the balanced voltage Va causes
charges Q\ and Q2 per unit length on these wires in the length dx.
These charges are affected by the presence of other wires in the length
dx and they are usually unbalanced. There will be equal and opposite

or balanced charges ± — ^ on each wire and unbalanced equal

charges on each w^re. Since the direct crosstalk is defined

as the effect of balanced charges and currents, only the balanced
charges should be considered in computing Ve- Letting Qa = ^
or the balanced charge on wire one per unit length, then:

Vtt ^^ > a^a i a^a^ai

where Ca is equal to the "transmission capacitance" per mile, i.e.,
the capacitance used in calculating «„ the attenuation constant and
Za the characteristic impedance of circuit a.

The above expression for Qa includes the reaction of charges in the
disturbed circuit. This reaction should not theoretically, be included
at this time, since, for convenience in calculation, the disturbed circuit
is assumed to have the impressed voltages Vm and Ve but no crosstalk
currents or charges as yet. The effect on Qa of charges in the disturbed
circuit, is, however, usually small compared with the effect of charges
in various tertiary circuits.

Designating the conductors of circuit & as 3 and 4, Ve is the difference
of the potentials of the electric field at 3 and 4 caused by the balanced
charges per unit length on 1 and 2. Therefore:

where pis, etc., are the potential coefficients.

For c.g.s. elst. units, pis = 2 log — where 513 and ru are the distances
indicated by Fig. 32. Therefore:

Ve = VaC„{pl3 - P23 " pH + p2^) = VaCapab-

The capacitance Ca may be obtained from measurements on a short
length of a multi-wire line. Its value is, however, only a few per cent

OP EN -WIRE CROSSTALK

227

greater than C„' the value for a single pair line (without capacitance
at the insulators). For a single pair having like wires in a horizontal

O TD o

IMAGE WIRES
Fig. 32 — Distances used in computing potential coefficients.

plane, CJ is readily calculated as follows:

where pn in c.g.s. elst. units is:

2{pii — pn) '

2 log

Sn
>n

The distances ^u and rn are indicated on Fig. 32.
The expression for Vc may be written:

Cn . C„

Ve = VaC„pab = VaCa paby^f — VuT ab J^ '

The coefficient Tab is called the "voltage transfer coefficient." It
is readily computed since it is a function of potential coefficients and
it is independent of the system of units used in computation. Since
Ca is about equal to Ca, Tab is about equal to the ratio of Ve to Va.

The shunt voltage Ve drives a current through the shunt admittance
of circuit b in the infinitesimal length dx of Fig. 31. This shunt
admittance is (Gb + jcoCb)dx which is very nearly equal to juiCbdx
where Cb is the transmission capacitance per mile of circuit h and

228 BELL SYSTEM TECHNICAL JOURNAL

CO = lirf where / is the frequency in cycles per second. This current
divides equally between the two ends of circuit h. The near-end
current is:

1 Ve

2 1 _^Z,

jwCbdx 2

The near-end direct crosstalk coefficient due to the electric field of
circuit a may be called Ne and is the limiting value of the following
expression as dx approaches zero :

la ' Kdx 2KIa " 1 , ^6 , '

y-^ +-Ydx

where K is the frequency in kilocycles. The near-end direct crosstalk
coefficient for the electric field is, therefore:

K\) IMc 2i^/„ 2K ' Ca'

The far-end current due to Ve of Fig. 31 is — ie and, therefore,
the far-end coefficient due to the electric field is — TV^.

The near-end and far-end crosstalk currents of Fig. 31 due to the
magnetic field are alike and are designated im which may be calculated
as follows:

. _ Vrn^ _ _ IgjcoMabdx
'"^ ~ 2Z, ~ 2Z,

The near-end or far-end crosstalk coefficients for the magnetic
field may be called Nm and Fm. They are alike and equal to the limit
of:

in. 10« , ,

-^ • v^-v- as ax approaches zero.
la Kdx

Therefore :

_ _ joiMai _ jirMal,

In the above, Mah is the mutual inductance per unit length between
circuits a and h. It is calculated in the same manner as p^h used in
computing V e- These methods of computing V e and F„, from the
distances rn, ru, Sn, etc., of Fig. 32 are not precise but are sufficiently
accurate for open-wire circuits since the diameters of the wires are

OPEN-WIRE CROSSTALK 229

small compared with their interaxial distances. The "image" wires
of Fig. 32 should, theoretically, be located farther below the equivalent
ground plane for calculations of mutual inductance. This alters Su,
etc. Since the distances between wires are small compared to those
between wires and images, the values of 5 are all about equal and have
practically no effect on the value of pab-

Therefore, Mah in c.g.s. elmg. units may be assumed numerically
equal to pah or:

Mah = pah = -TT-, '

In c.g.s. elst. units CJ = wn ;r^ which is also the expression

2(^11 - P12)

for XjLa' in c.g.s. elmg. units where LJ is the external inductance of

circuit a, i.e., the inductance due to the magnetic field external to the

wires of circuit a. Therefore :

Mah = TabLa ,

where Mah and La may be expressed in any system of units.

The near-end or far-end direct coeflficient for the magnetic field
may, therefore, be written :

(2) N^ = F„,= - hl^K 109.

The above expression is almost equal to Ne, the near-end coefficient
for the electric field.
It may be written:

Nm = N.

jcoLa'jcoCa'
ZajwCaZbjcoCb

Now ZajcoCa is vcry nearly equal to Za(Ga + jo^Ca) which is ja- Like-
wise ZbjwCb is very nearly equal to 76. If the circuits had no resistance
or leakance the propagation constant would be 70 — jw^LJCa' or
joijv where v is the speed of light in miles per second. Therefore:

AT ^^TO^ 1

JMm = very nearly.

7a76

The total direct crosstalk coefficients are:

(3) Na = Nr, + N. = nA\-^^\ = 2N, approx.

\ 7«76 /

At carrier frequencies the ratio of 70 to 7,, (or 7/,) is about equal to

230 BELL SYSTEM TECHNICAL JOURNAL

the ratio of the actual speed of propagation to the speed of light, i.e.,
180,000 to 186,000 or about .97. Therefore ( 1 + ^ ) is about 1.94.

\ TaTb /

(4) Fa = N„.- N, = Nei—- A

\ TaTb /

AT- / n/C I • 90 Ofn + at \

= A^e I - .06 + 7 — — ^ — j approx.

The attenuation of the disturbing and disturbed circuits may not be
neglected in evaluating the expression ( — ^- 1 ) •

\ 7a76 /

The expression given for Ne in equation (1) above may be written:

^aTahW Ca Cft

N.= -

2K Ca Ca

This assumes ZnjcoCa equals ja. At carrier frequencies 7a is about
equal to jl3a which is about JTrK/90 since the speed of propagation is
about 180,000 miles per second. The expression for Ne may, therefore,
be written in the following simple approximate form:

_ _ . TrrgfelO'^ Ca Cb
^ " ~ ^ 180 Ca' 'Ca'

The ratio of C„ to C„' does not ordinarily exceed 1.02. For like
circuits, therefore:

.. - jirTatW
' = 180 approx-

On Fig. S3 the magnitudes of Nd and Fd are plotted against frequency
for 8-inch spaced conductors .128-inch in diameter. Both Fd and A^^
are divided by Tab to make the curves applicable to any circuit combi-
nation. These curves show that Na is practically independent of
frequency (above a few hundred cycles) but Fd decreases rapidly with
frequency for several thousand cycles.

Indirect Crosstalk Coefficients
Expressions for the indirect crosstalk coefficients used in computing
the indirect component of transverse crosstalk coupling will now be
derived. The derivation first covers the case of a single representative
tertiary circuit. Fig. 34 shows a thin transverse slice of the parallel
of the three circuits, the thickness of the slice being the infinitesimal
length dx. The only tertiary circuit to be considered for the present
is the metallic circuit composed of wires 5 and 6 and designated as c.
There are other possible tertiary circuits in the system of 6 wires.

OPEN-WIRE CROSSTALK

231

for example, the phantom circuit composed of wires 1 and 2 as one
side and 5 and 6 as the other side. The method of estimating the
total effect of all possible tertiary circuits will be discussed later.

40,000

35,000

■

Tab

15,000

10,000

—

0.1 0.2 0.3 0.4 0.5 I 2 3 4 5 10

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 33 — Variation of direct crosstalk coefficients with frequency.

The immediate problem is to compute the crosstalk currents In
circuit b at the ends of the length dx due to currents and charges in
circuit c in this length and caused by transmission over circuit a
through dx.

The crosstalk currents in circuit b due to currents and charges in
circuit a were computed by determining the equivalent series and
shunt e.m.f.'s in circuit b. The effect of currents and charges in
circuit c on crosstalk currents in circuit b may be computed in a similar
manner. The series e.m.f. in circuit b proportional to the current in

232

BELL SYSTEM TECHNICAL JOURNAL

circuit c will, however, be negligible compared with the series e.m.f.
proportional to current in circuit a. This is evident since the current
in circuit c is a crosstalk current which approaches zero as dx ap-
proaches zero while the current in circuit a does not vary with dx.

ICL

Fig. 34 — Schematic used in deriving formulas for indirect crosstalk coefficients.

The shunt e.m.f. in circuit h dependent on the charges of circuit c is
not, however, negligible compared with the shunt e.m.f. in circuit h
due to charges in circuit a since the charges in both a and c approach
zero as dx decreases. In other words the magnetic field of circuit c
may be neglected but the electric field must be considered. (Both
fields must be considered in computing interaction crosstalk.)

To determine the equivalent shunt e.m.f. in circuit h which depends
upon the electric field of circuit c the voltage between the wires of
circuit c must be determined. If circuit c did not exist, the electric
field of circuit a would cause a difference of potential between the
points actually occupied by wires 5 and 6 at the left-hand end of dx
in Fig. 34. This difference of potential would be:

V ac ^ V a^a yac ' a-l ac

With circuit c present, this difference of potential is changed to
Vc, the actual voltage across circuit c. The voltage could not change
from Vac to Vc without charges on circuit c and the charge per wire
per unit length is proportional to the change in voltage from V,,,- to
Vc which may be designated Uc. The equivalent shunt e.m.f. in

OP EN -WIRE CROSSTALK 233

circuit h due to the presence of charges in circuit c is, therefore, pro-
portional to Uc.
By definition :

Vac + Uc= Vc or Uc= Vc- Vac.

Since the crosstalk current in circuit c approaches zero as dx ap-
proaches zero, Vc must also approach zero and Uc approaches — Vac-
The shunt e.m.f. in circuit h due to charges on circuit a was computed
as:

Ve = VaTa^%'

To allow for the electric field of circuit c, Ve must be augmented by:

Cc Cc Cc

V e ^ U c-i cb '7^1 ^^ VacJ- c&~7^ ^ Va-I- aci cb yr~f '

Cc »-c Cc

Since the part of the direct near-end crosstalk coefficient resulting

from Ve was found to be iV^ = - jirZaTabCb^O^ ^n . by proportion

the indirect near-end coefficient resulting from VJ will be:

(5) Ni = jirZaTacTcbCblO' ^ = ^^^^m'^^' ^PP''^^-

Since the far-end crosstalk current resulting from a shunt voltage in
circuit b is opposite in sign to the near-end current, the indirect far-end
coefficient will be:

(6) Fi= - Ni.

Total Crosstalk Coefficients
The total near-end and far-end crosstalk coefficients used in com-
puting transverse crosstalk coupling will be the sum of the direct and
indirect coefficients or:

(7) N = Na + Ni.

(8) F = Fd + Fi = Fd- Ni.

The expressions for Fi and Ni are about independent of frequency
in the carrier-frequency range because Za does not depend much on
frequency above a few thousand cycles, Cb is about independent of
frequency and TacTcb depends only on the cross-sectional dimensions
of the wire configuration.

Since, as indicated by Fig. 2>2>, Nd is usually about independent of

234 BELL SYSTEM TECHNICAL JOURNAL

frequency and since N = Nd -\- Ni is largely determined by Nd, the
near-end coefficient N is about independent of frequency above a
few hundred cycles. The far-end coefficient F is about independent
of frequency above a few thousand cycles where it is largely determined
by Fi.

The preceding discussion of indirect crosstalk coefficients covered
only the effect of charges in the single metallic tertiary circuit c of
Fig. 34. The indirect coefficient in a practical case may be estimated
with fair accuracy by considering all the more important tertiary
circuits in a similar manner. It was shown that the final voltage of
tertiary circuit c was zero. Similarly, the final voltage of each tertiary
circuit is zero. This includes any tertiary circuits involving the two
wires of the disturbing circuit in multiple. The average voltage of
the two wires of the disturbing circuit is zero and the voltage across
the disturbing circuit is balanced. As previously stated, this voltage
does not become unbalanced as a result of transverse crosstalk in any
infinitesimal length but it may become unbalanced due to interaction
crosstalk.

The charges per unit length on the various tertiary circuits are the
same as those which would be caused by impressing a system of
voltages equal and opposite to those induced by the balanced charges
per unit length which would be on the two wires of the disturbing
circuit if this circuit were the only pair on the line. Assuming such
a system of impressed voltages, it is not practicable to accurately
compute the charges in any tertiary circuit since this depends on the
voltages impressed on all the tertiary circuits and the couplings
between the various tertiary circuits. Advantage may be taken,
however, of the fact that the charge on a tertiary circuit will depend
mostly on the voltage impressed on that circuit provided it is not
heavily coupled with other circuits.

It is possible to divide the various voltages impressed on the tertiary
circuits into components such that (1) equal voltages are impressed
on wires of a "ghost" circuit composed of all the wires on the line
with ground return, (2) balanced voltages are impressed on each pair
used for transmission purposes (except the disturbed and disturbing
circuits) and (3) balanced voltages are impressed on each possible
phantom of two pairs used for transmission purposes.

Such a system of impressed voltages and tertiary circuits is con-
venient for computation since the charge on any tertiary circuit
largely depends on the voltage impressed on that circuit. If accurate
calculations of the charges were practicable, a simpler system of
tertiary circuits could be used to obtain the same final result, i.e.,

OPEN-WIRE CROSSTALK 235

single-wire tertiary circuits with ground return could be used. Com-
putation with such tertiary circuits is impracticable because of the
large coupling between them.

In the elaborate system of tertiary circuits described above, the
ghost circuit may be neglected. The voltage impressed across this
circuit is the average of all the voltages impressed on the various
wires. These voltages may be plus or minus and the average tends
to be small. Also, the charge per pair per impressed volt is usually
much less for the ghost circuit than for a phantom circuit due to the
relatively small capacitance between a pair and ground as compared
with that between two pairs.

The pairs used for transmission purposes may usually be disregarded,
also, since their coupling with the disturbing and disturbed circuits is
much smaller than that of the phantom tertiary circuits.

The practical method of computing the indirect crosstalk coefficient
is, therefore, to consider as tertiary circuits a considerable number of
phantoms composed of pairs used for transmission purposes including
the disturbed and disturbing pairs. In calculating the charge in any
tertiary circuit, the voltages impressed on other tertiary circuits are
disregarded.

In calculating the effect of a single tertiary circuit c, the expression
for the indirect coefficient contained the factor TacTcb- To estimate
the effect of all the tertiary circuits, this factor should be replaced by :

— ZLTapTph.

This expression assumes that there are n pairs on the line and that
w — 2 of these pairs are close enough to the disturbing and disturbed
pairs to appreciably affect the indirect crosstalk between them. The
subscript p indicates any phantom of the m pairs including the dis-
turbed and disturbing pairs. The summation is for all possible
phantoms each consisting of two of the ni pairs. If the voltages
induced by the balanced charges Q,,' of pair a are Vr and Vs for the
two sides of a phantom, the balanced voltage assumed to be impressed

2
across the phantom is— (F^ — Vr). Other parts of Vr and Vs are

used in the "ghost" voltage and in balanced voltages across other
phantoms.

Tap and Tpb are voltage transfer coefficients relating balanced
impressed voltage on the disturbing circuit to induced voltage on the
disturbed circuit. Tap involves C/ the transmission capacitance of
circuit a on a single pair line. Tph involves the transmission capaci-

236

BELL SYSTEM TECHNICAL JOURNAL

tance of a particular phantom on a line having only that phantom
present. This capacitance is the ratio of balanced charge (on each
side of the phantom) to the balanced impressed voltage. The phantom
capacitance may be readily estimated from the potential coefficients.
For example, if the phantom involves pairs 1-2 and 5-6 the phantom
capacitance is very nearly:

C =

Pn + ^22 + ^55 + ^66 + 2^12 + 2^56 " 2^i5 — 2/)25 — 2pi& — 2/?2G

If the disturbing circuit is pair 1-2 and the disturbed circuit is
pair 3-4:

T =

Plh + Pu — p2h — p2i

2{pn — pn)

Tpb = ~~ {plZ + p2S + Pib + Pi6 — Pu — p2i — p3b — Pse)-

These computations of indirect coefficients are necessarily laborious.
They can be simplified to some extent by ignoring phantoms for which
either Tap or Tpb is zero or small. For example, the voltage transfer
coefficient is zero for pair 1-2 to such phantoms as 1-2 and 11-12,
11-12 and 21-22, etc.

In the following table are given comparisons of far-end crosstalk
coefficients as measured in a 40-wire line and as computed by the
methods discussed above. The spacing of the various wires and
crossarms is indicated by Fig. 27A. The measured values are for
40 wires and the computed values are for 10, 20 and 30 wires. It
will be seen that a considerable number of wires must be taken into
account in the computations in order to obtain a fair check with the
coefficient measured for a heavy line.

Far-End Crosstalk Per Mile Per Kilocycle

Combination

1-2 to 3-4

1-2 to 11-12

1-2 to 9-10

Computed for 10 wires

45
63
69

28
47
58
70

Computed for 20 wires

Computed for 30 wires

Measured for 40 wires

OPEN- WIRE CROSSTALK

237

Interaction Crosstalk Coefficient
It was assumed in the discussion of crosstalk coefficients that the
"interaction crosstalk coefficient" NacN ch^Q"^ was nearly equal to
— IFijclK. This relation is deduced below, for a representative
tertiary circuit c, from the expressions for Fi and Nd given by equations
(3) and (6) above. Nac may be obtained by using the expressions
for Ne and Nd given by equations (1) and (3) above. In these equa-
tions, subscript c should be substituted for subscript b. The expression
for Nac becomes:

Nac = - jirZaTacCcW^,

1 +

Jajc

Deriving a similar approximate expression for Net:

F- C
N N JO-s = _ i_L ^ JrfL

1 +

_7ol

laic

1 +

70^

Iclh

This assumes Zcj<JiCc = Zc(Gc + jirCc) which is 7c.
Crosstalk measurements indicate that the ratio of 70 to 7a or 7b or
7c is about .97 at carrier frequencies and CajCJ about 1 .02. Therefore :

NacNcb\0-' =

Fijc
2K

1.02(1.94)2 = - 2Fi7c/i^ approx.

The above discussion covers the case of a single tertiary circuit c.
In the practical case the known crosstalk coefficient Fi includes the
effect of a large number of tertiary circuits which have various values
of 7c. Obtaining the interaction crosstalk coefficient from the ex-
pression — IFadK involves assuming a representative value of 7,..
At carrier frequencies 7c is about equal to j^c which for all important
tertiary circuits is in the neighborhood of JTrK/90.

A known value of the crosstalk coefficient Fi expresses the effects
of the electric fields of many tertiary circuits in one infinitesimal slice
and includes the alteration of the field of any one tertiary circuit due
to the presence of the others. The interaction crosstalk coefficient
involves consideration of both electric and magnetic fields. In any
one slice, the electric field of a tertiary circuit determines its crosstalk
into the disturbed circuit but the current in the tertiary circuit at the
end of the slice is determined by both the electric and magnetic fields
of the disturbing circuit. The tertiary circuit current is transmitted
into another slice and sets up electric and magnetic fields which both
contribute to the crosstalk current in the disturbed circuit.

238 BELL SYSTEM TECHNICAL JOURNAL

In the case of a single tertiary circuit, the electric and magnetic
effects expressed by the interaction coefficient are simply related and
the interaction coefficient is simply related to Fi. With many tertiary
circuits the relation between the electric and magnetic fields of any
one tertiary circuit is altered by the presence of the others and the
relative importance of electric and magnetic fields in the interaction
effect varies among the tertiary circuits (except for the ideal case of
" non-dissipative " circuits). In a practical case, therefore, the inter-
action coefficient is only approximately proportional to F,. Measure-
ments of the interaction coefificients by indirect methods have, however,
indicated that the approximation is satisfactory for purposes of
practical transposition design.

Symposium on Wire Transmission of Symphonic Music
and Its Reproduction in Auditory Perspective

Basic Requirements*

By HARVEY FLETCHER

The fundamental requirements involved In a system capable of picking
up orchestral music, transmitting it a long distance, and reproducing it in a
large hall, are discussed in this paper.

IN THIS electrical era one is not surprised to hear that orches-
tral music can be picked up in one city, transmitted a long
distance, and reproduced in another. Indeed, most people think
such things are commonplace. They are heard every night on the
radio. However, anyone who appreciates good music would not
admit that listening even to the best radio gives the emotional thrill
experienced in the concert hall. Nor is it evident that a listener in a
small room ever will be able to get the same effect as that experienced
in a large hall, although it must be admitted that such a question is
debatable. The proper answer will involve more than a consideration
of only the physical factors.

This symposium describes principles and apparatus involved in the
reproduction of music in large halls, the reproduction being of a character
that may give even greater emotional thrills to music lovers than those
experienced from the original music. This statement is based upon
the testimony of those who have heard some of the few concerts
reproduced by the apparatus which will be described in the papers of
this symposium.

It is well known that when an orchestra plays, vibrations which are
continually changing in form are produced in the air of the concert
hall where the orchestra is located. An ideal transmission and
reproducing system may be considered as one that produces a similar
set of vibrations in a distant concert hall in which is executed the same
time-sequence of changes that takes place in the original hall. Since
such changes are different at different positions in the hall, the use of
such an ideal system implies that at corresponding positions in the

* First paper in the Symposium. Presented at Winter Convention of A. I. E. E.,
New York City, Jan. 23-26, 1934. Published in Electrical Engineering, January,
1934.

239

240 BELL SYSTEM TECHNICAL JOURNAL

two halls this time-sequence should be the same. Obviously, this
never can be true at every position unless the halls are the same size
and shape; corresponding positions would not otherwise exist. Let
us consider the case where the two halls are the same size and shape
and also have the same acoustical properties. Let us designate the
first hall in which the music originates by 0, and the second one in
which the music is reproduced by R. What requirements are necessary
to obtain perfect reproduction from 0 into R such that any listener
in any part of R will receive the same sound effects as if he were in the
corresponding position in O?

Suppose there were interposed between the orchestra and the audi-
ence a flexible curtain of such a nature that it did not interfere with a
free passage of the sound, and which at the same time had scattered
uniformly over it microphones which would pick up the sound waves
and produce a faithful electrical copy of them. Assume each micro-
phone to be connected with a perfect transmission line which termi-
nates in a projector occupying a corresponding position on a similar
curtain in hall R. By a perfect transmission line is meant one that
delivers to the projector electrical energy equal both in form and
magnitude to that which it receives from the microphone. If these
sound projectors faithfully transform the electrical vibrations into
sound vibrations, the audience in hall R should obtain the same effect
as those listening to the original music in hall 0.

Theoretically, there should be an infinite number of such ideal sets
of microphones and sound projectors, and each one should be infin-
itesimally small. Practically, however, when the audience is at a
considerable distance from the orchestra, as usually is the case, only
a few of these sets are needed to give good auditory perspective; that
is, to give depth and a sense of extensiveness to the source of the
music. The arrangement of some of these simple systems, together
with their effect upon listeners in various parts of the hall, is described
in the paper by Steinberg and Snow (page 245).

In any practical system it is important to know how near these ideal
requirements one must approach before the listener will be aware that
there has been any degradation from the ideal. For example, it is
well known that whenever a sound is suddenly stopped or started, the
frequency band required to transmit the change faithfully is infinitely
wide. Theoretically, then, in order to fulfill these ideal requirements for
transmitting such sounds, all three elements in the transmission system
should transmit all possible frequencies without change. Practically,
because of the limitations of hearing, this is not necessary. If the
intensities of some of the component frequencies required to represent

BASIC REQUIREMENTS 241

such a change are below the threshold of audibility it is obvious that
their elimination will not be detected by the average normal ear.
Consequently, for highgrade reproduction of sounds it is obvious that,
except in very special cases, the range of frequencies that the system
must transmit is determined by the range of hearing rather than by
the kind of sound that is being reproduced.

Tests have indicated that, for those having normal hearing, pure
tones ranging in frequency from 20 to 20,000 cycles per second can be
heard. In order to sense the sounds at either of these extreme limits,
they must have very high intensity. In music these frequencies
usually are at such low intensities that the elimination of frequencies
below 40 c.p.s. and those above 15,000 c.p.s. produces no detectable
difference in the reproduction of symphonic music. These same tests
also indicated that the further elimination of frequencies beyond either
of these limits did begin to produce noticeable effects, particularly on
a certain class of sounds produced in the orchestra. For example, the
elimination of all frequencies above 13,000 c.p.s. produced a detectable
change in the reproduced sound of the snare drum, cymbals, and
castanets. Also, the elimination of frequencies below 40 c.p.s.
produced detectable differences in reproduced music of the base viol,
the bass tuba, and particularly of the organ.

Within this range of frequencies the system (the combination of the
microphone, transmission line, and loud speaker) should reproduce the
various frequencies with the same efficiency. Such a general statement
sounds correct, but a careful analysis of it would reveal that when any
one tried to build such a system or tried to meet such a requirement
he would have great difficulty in understanding what it meant.

For example, for reproducing all the frequencies within this band,
a certain system may be said to have a uniform efficiency when it
operates between two rooms under the condition that the pressure
variation at a certain distance away from the sound projector is the
same as the pressure variation at a certain position in front of the
microphone. It is obvious, however, that in other positions in the two
rooms this relation would not in general hold. Also, if the system
were transferred into another pair of rooms the situation would be
entirely changed. These difficulties and the way they were met are
discussed in the papers of this symposium that deal with loud speakers
and microphones (p. 259) and with methods of applying the reproducing
system to the concert hall (p. 301). It will be obvious from these
papers that the criterion for determining the ideal frequency charac-
teristics to be given to the system is arbitrary within certain limits.
However, solving the problem according to criteria adopted produced
a system that gave very satisfactory results.

242 BELL SYSTEM TECHNICAL JOURNAL

Besides the requirement on frequency response just discussed, the
system also must be capable of handling sound powers that vary
through a very wide range. If this discussion were limited to the type
of symphonic music that now is produced by the large orchestras, this
range would be about 10,000,000 to 1, or 70 decibels. To reproduce
such music then, the system should be capable of handling the smallest
amount of power without introducing extraneous noises approaching
it in intensity, and also reproduce the most intense sounds without
overloading any part of the transmission system. However, this range
is determined by the capacities of the musical instruments now avail-
able and the man power that conveniently can be grouped together
under one conductor. As soon as a system was built that was capable
of handling a much wider range, the musicians immediately took
advantage of it to produce certain effects that they previously had
tried to obtain with the orchestra alone, but without success because
of the limited power of the instruments themselves. For these
reasons it seems clear that the desirable requirements for intensity
range, as well as those for frequency range, are determined largely by
the ear rather than by the physical characteristic of any sound. An
ideal transmission should, without introducing an extraneous audible
sound, be capable of reproducing a sound as faintly as the ear can hear
and as loudly as the ear can tolerate. Such a range has been deter-
mined with the average normal ear when using pure tones. The
results of recent tests are shown in Fig. 1.

The ordinates are given in decibels above the reference intensity
which is 10~'^ watts per square centimeter. The values are for field
intensities existing in an air space free from reflecting walls. The
most intense peaks in music come in the range between 200 and 1000
c.p.s. Taking an average for this range it may be seen that there is
approximately a 100-db range in intensity for the music, provided
about 10 db is allowed for the masking of sound in the concert hall
even when the audience is quietest.

The music from the largest orchestra utilizes only 70 db of this
range when it plays in a concert hall of usual size. To utilize the full
capabilities of the hearing range the ideal transmission system should
add about 10 db on the pp side and 20 db on the^ side of the range.
The capacity of the sound projectors necessary to reach the maximum
allowable sound that the ear can tolerate varies with the size of the
room. A good estimate can be obtained by the following consider-
ation.

If T is the time of reverberation of the hall in seconds, E the power
of the sound source in watts, / the maximum energy density per cubic

BASIC REQUIREMENTS

243

centimeter in joules, and T the volume of the hail in cubic centimeters,
then it is well known that

/ =

V '

(1)

6 log« 10

Measurements have shown that when the sound intensity in a free
field reaches about 10^^ watts per square centimeter, the average
person begins to feel the sound. This maximum value is approxi-

:;,

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/■

1 1

'large
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100 500 1000 5000

FREQUENCY IN CYCLES PER SECOND

10,000

50,000

Fig. 1 — Limits of audible sound as determined by recent tests.

mately the same for all frequencies in the important audible range.
Any higher intensities, and for some persons somewhat lower inten-
sities, become painful and may injure the hearing mechanism. This
intensity corresponds to an energy density / of 3 X 10~^ joules. Using
this figure as the upper limit to be tolerated by the human ear, then,
the maximum power of the sound source must be given by

£ = 4.1 X 10-8^.

(2)

For halls like the Academy of Music in Philadelphia and Carnegie
Hall in New York City, in which the volume I^ is approximately
2 X 10"* cubic centimeters and the reverberation time about 2 seconds.

244 BELL SYSTEM TECHNICAL JOURNAL

E, the power of the sound source, is approximately 400 watts. For
other halls it may be seen that the power required for this source is
proportional to the volume of the hall and inversely proportional to
the reverberation time. A person would experience the sense of
feeling when closer than about 10 meters to such a source of 400 watts
power, even in free open space. Hence it would be unwise to have
seats closer than 10 or 15 meters from the stage when such powers are
to be used.

These, then, are the general fundamental requirements for an ideal
transmission system. How near they can be realized with apparatus
that we now know how to build will be discussed in the papers included
in this symposium.

A system approximately fulfilling these requirements was con-
structed and used to reproduce the music played by the Philadelphia
Orchestra. The first public demonstration was given in Constitution
Hall, Washington, D. C, on the evening of April 27, 1933, under the
auspices of the National Academy of Sciences. At that time, Dr.
Stokowski, Director of the Philadelphia Orchestra, manipulated the
electric controls from a box in the rear of Constitution Hall while the
orchestra, led by Associate Conductor Smallens, played in the Academy
of Music in Philadelphia.

Three microphones of the type described in the paper by Wente and
Thuras (p. 259) were placed before the orchestra in Philadelphia, one on
each side and one in the center at about 20 feet in front of and 10 feet
above the first row of instruments in the orchestra. The electrical
vibrations generated in each of these microphones were amplied by
voltage amplifiers and then fed into a transmission line which was
extended to Washington by means of telephone cable. The con-
struction of these lines, the equipment used with them, and their
electrical properties, are described in the paper by Affel, Chesnut, and
Mills (p. 285). In Constitution Hall at Washington, D. C, these
transmission lines were connected to power amplifiers. The type of
power amplifiers and voltage amplifiers used are described in the paper
by Scriven (p. 278). The output of these amplifiers fed three sets of
loud speakers like those described in the paper by Wente and Thuras.
They were placed on the stage in Constitution Hall in positions cor-
responding to the microphones in the Academy of Music, Philadelphia.

Judging from the expression of those who heard this concert, the
development of this system has opened many new possibilities for the
reproduction and transmission of music that will create even a greater
emotional appeal than that obtained when listening to the music
coming directly from the orchestra through the air.

Physical Factors*

By J. C. STEINBERG and W. B. SNOW

In considering the physical factors affecting it, auditory perspective is
defined in this paper as being reproduction which preserves the spatial rela-
tionships of the original sounds. Ideally, this would require an infinite
number of separate microphone-to-speaker channels; practically, it is shown
that good auditory perspective can be obtained with only 2 or 3 channels.

ABILITY to localize the direction, and to form some judgment of
the distance from a sound source under ordinary conditions of
listening, are matters of common experience. Because of this faculty
an audience, when listening directly to an orchestral production, senses
the spatial relations of the instruments of the orchestra. This spatial
character of the sounds gives to the music a sense of depth and of
extensiveness, and for perfect reproduction should be preserved. In
other words, the sounds should be reproduced in true auditory per-
spective.

In the ordinary methods of reproduction, where only a single loud
speaking system is used, the spatial character of the original sound is
imperfectly preserved. Some of the depth properties of the original
sound may be conveyed by such a system, ^ but the directional proper-
ties are lost because the audience tends to localize the sound as coming
from the direction of a single source, the loud speaker. Ideally, there
are two ways of reproducing sounds in true auditory perspective. One
is binaural reproduction which aims to reproduce in a distant listener's
ears, by means of head receivers, exact copies of the sound vibrations
that would exist in his ears if he were listening directly. The other
method, which was described in the first paper of this series, uses loud
speakers and aims to reproduce in a distant hall an exact copy of the
pattern of sound vibration that exists in the original hall. In the
limit, an infinite number of microphones and loud speakers of infin-
itesimal dimensions would be needed.

Far less ideal arrangements, consisting of as few as two microphone-
loudspeaker sets, have been found to give good auditory perspective.
Hence, it is not necessary to reproduce in the distant hall an exact
copy of the vibrations existing in the original hall. What physical

* Second paper in the Symposium on Wire Transmission of Symphonic Music
and Its Reproduction in Auditory Perspective. Presented at Winter Convention of
A. I. E. E., New York City, Jan. 23-26, 1934. Published in FJeclrical Eni^ineering,
January, 1934.

245

246 BELL SYSTEM TECHNICAL JOURNAL

properties of the waves must be preserved then, and how are these
properties preserved by various arrangements of 2- and 3-channel
loudspeaker reproducing systems? To answer these questions, some
very simple localization tests have been made with such systems.
Perhaps attention can be focused more easily on their important
properties by considering briefly the results of these tests.

Localization Afforded by Multichannel Systems

In Fig. 1 is shown a diagram of the experimental set-up that was
used. The microphones, designated as LM (left), CM (center), and
Rj\I (right), were set on a "pick-up" stage that was marked out on
the floor of an acoustically treated room. The loud speakers, desig-
nated as LS, CS, and RS, were placed in the front end of the auditorium
at the Bell Telephone Laboratories and were concealed from view by a
curtain of theatrical gauze. The average position of a group of twelve
observers is indicated by the cross in the rear center part of the
auditorium.

The object of the tests was to determine how a caller's position on
the pick-up stage compared with his apparent position as judged by
the group of observers in the auditorium listening to the reproduced
speech. Words were uttered from some 15 positions on the pick-up
stage in random order. The 9 positions shown in Fig. 1 were always
included in the 15, the remaining positions being introduced to mini-
mize memory effects. The reproducing system was switched off while
the caller moved from one position to the other.

In the first series of tests, the majority of the observers had no
previous experience with the set-up. They simply were given a sheet
of coordinate paper with a single line ruled on it to indicate the line
of the gauze curtain and asked to locate the apparent position of the
caller with respect to this line. Following these tests, the observers
were permitted to listen to speech from various announced positions
on the pick-up stage. This gave them some notion of the approximate
outline of what might be called the "virtual" stage. These tests then
were repeated. As there was no significant difference in results, the
data from both tests have been averaged and are shown in Fig. 1.

The small diagram at the top of Fig. 1 shows the caller's positions
with respect to the microphone positions on the pick-up stage. The
corresponding average apparent positions when reproduced are shown
with respect to the curtain line and the loudspeaker positions. The
type of reproduction is indicated symbolically to the right of the
apparent position diagrams.

With 3-channel reproduction there is a reasonably good corre-

PHYSICAL FACTORS

247

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248 BELL SYSTEM TECHNICAL JOURNAL

spondence between the caller's actual position on the pick-up stage
and his apparent position on the virtual stage. Apparent positions to
the right or left correspond with actual positions to the right or left,
and apparent front and rear positions correspond with actual front
and rear positions. Thus the system afforded lateral or "angular"
localization as well as fore and aft or "depth" localization. For
comparison, there is shown in the last diagram the localization afforded
by direct listening. The crosses indicate a caller's position in back
of the gauze curtain and the circles indicate his apparent position as
judged by the observers listening to his speech directly. In both
cases, as the caller moved back in a straight line on the left or right side
of the stage, he appeared to follow a curved path pulling in toward the
rear center; e.g., compare the caller positions 1, 2, 3, with the apparent
positions 1, 2, 3. This distortion was somewhat greater for 3-channel
reproduction than for direct listening.

The results obtained with the 2-channel system show two marked
differences from those obtained with 3-channel reproduction. Posi-
tions on the center line of the pick-up stage (i.e., 4, 5, 6) all appear in
the rear center of the virtual stage, and the virtual stage depth for all
positions is reduced. The virtual stage width, however, is somewhat
greater than that obtained with 3-channel reproduction.

Bridging a third microphone across the 2-channel system had the
effect of pulling the center line positions 4, 5, 6, forward, but the
virtual stage depth remained substantially that afforded by 2-channel
reproduction, while the virtual stage width was decreased somewhat.
In this and the other bridged arrangements the bridging circuits
employed amplifiers, as represented by the arrows in Fig. 1, in such
a way that there was a path for speech current only in the indicated
direction.

Bridging a third loud speaker across the 2-channel system had the
effect of increasing the virtual stage depth and decreasing the virtual
stage width, but positions on the center line of the pick-up stage
appeared in the rear center of the virtual stage as in 2-channel repro-
duction.

Bridging both a third microphone and a third loud speaker across the
2-channel system had the effect of reducing greatly the virtual stage
width. The width could be restored by reducing the bridging gains,
but fading the bridged microphone out caused the front line of the
virtual stage to recede at the center, whereas fading the bridged loud
speaker out reduced the virtual stage depth. No fixed set of bridging
gains was found that would enable the arrangement to create the
virtual stage created by three independent channels. The gains used in

PHYSICAL FACTORS 249

obtaining the data shown in Fig. 1 are indicated at the right of the
symboHc circuit diagrams.

Factors Affecting Depth Localization
Before attempting to explain the results that have been given in the
foregoing, it may be of interest to consider certain additional observa-
tions that bear more specifically upon the factors that enter into the
"depth" and "angular" localization of sounds. The microphones on
the pick-up stage receive both direct and reverberant sound, the
latter being sound waves that have been reflected about the room in
which the pick-up stage is located. Similarly, the observer receives
the reproduced sounds directly and also as reverberant sound caused
by reflections about the room in which he listens. To determine the
efi^ects of these factors, the following three tests were made:

1. Caller remained stationary on the pick-up stage and close to
microphone, but the loudness of the sound received by the observer
was reduced by gain control. This was loudness change without a
change in ratio of direct to reverberant sound intensity.

2. Caller moved back from microphone, but gain was increased to
keep constant the loudness of the sound received by the observer.
This was a change in the ratio of direct to reverberant sound intensity
without a loudness change.

3. Caller moved back from microphone, but no changes were made
in the gain of the reproducing system. This changed both the ratio
and the loudness.

All of the observers agreed that the caller appeared definitely to recede
in all three cases. That is, either a reduction in loudness or a decrease
in ratio of direct to reverberant sound intensity, or both, caused the
sound to appear to move away from the observer. Position tests using
variable reverberation with a given pick-up stage outline showed that
increasing the reverberation moved the front line of the virtual stage
toward the rear, but had slight effect upon the rear line. When the
microphones were placed outdoors to eliminate reverberation, reducing
the loudness either by changing circuit gains or by increasing the
distance between caller and microphone moved the whole virtual
stage farther away. It is because of these effects that all center line
positions on the pick-up stage appeared at the rear of the virtual stage
for 2-channel reproduction.

It has not been found possible to put these relationships on a quan-
titative basis. Probably a given loudness change, or a given change in
ratio of direct to reverberant sound intensity, causes different sensa-
tions of depth depending upon the character of the reproduced sound

250 BELL SYSTEM TECHNICAL JOURNAL

and upon the observe-'s familiarity with the acoustic conditions sur-
rounding the reproduction. Since the depth locaHzation is inaccurate
even when Hstening directly, it is difficult to obtain sufficiently accurate
data to be of much use in a quantitative way. Because of this inac-
curacy, good auditory perspective may be obtained with reproduced
sounds even though the properties controlling depth localization depart
materially from those of the original sound.

Angular Localization

Fortunately, the properties entering into lateral or angular local-
ization permit more quantitative treatment. In dealing with angular
localization, it has been found convenient to neglect entirely the
effects of reverberant sound and to deal only with the properties of the
sound waves reaching the observer's ears without reflections. The
reflected waves or reverberant sounds do appear to have a small
effect on angular localization, but it has not been found possible to
deal with such sound in a quantitative way. One of the difficulties
is that, because of differences in the build-up times of the direct and
reflected sound waves, the amount of direct sound relative to rever-
berant sound reaching the observer's ears for impulsive sounds such
as speech and music is much greater than would be expected from
steady state methods of dealing with reverberant sound.

For the case of a plane progressive wave from a single sound source,
and where the observer's head is held in a fixed position, there are
apparently only three factors that can assist in angular localization:
namely, phase difference, loudness difference, and quality difference
between the sounds received by the two ears.

In applying these factors to the localization of sounds from more
than one source, as in the present case, the effects of phase differences
have been neglected. It is difficult to see how phase differences in
this case can assist in localization in the ordinary way. The two re-
maining factors, loudness and quality differences, both arise from the
directivity of hearing. This directivity probably is due in part to the
shadow and diffraction effects of the head and to the differences in the
angle subtended by the ear openings. Measurements of the directivity
with a source of pure tone located in various positions around the
head in a horizontal plane have been reported by Sivian and White.^
From these measurements, the loudness level differences between near
and far ears have been determined for various frequencies. These
differences are shown in Fig. 2 from which, using the pure tone data
given, similar loudness level differences for complex tones may be
calculated. Such calculated differences for speech are shown in Fig. 3.

PHYSICAL FACTORS

251

300~

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B- FAR EAR

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ANGLE IN DEGREES

Fig.

2 — Variation in loudness level as a sound source is rotated in a horizontal plane
around the head.

252

BELL SYSTEM TECHNICAL JOURNAL

0"^

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Fig. 3 — Variation in loudness as a speech source is rotated in a horizontal plane

around the head.

As may be inferred from the varying shapes of the curves of Fig. 2,
the directive effects of hearing introduce a frequency distortion more
or less characteristic of the direction from which the sound comes.
Thus the character or quaUty of complex sounds varies with the angle
of the source. There are quality differences at each ear for various
angles of source, and quality differences between the two ears for a
given angle of source. In Fig. 4 is shown the frequency distortion at
the right ear when a source of sound is moved from a position on the
right to one on the left of an observer. It is a graph of the "difference"
values of Fig. 2 for an angle of 90 degrees. Frequencies above 4,000
cycles per second are reduced by as much as 15 to 30 decibels. This
amount of distortion is sufficient to affect materially the quality of
speech, particularly as regards the loudness of the sibilant sounds.

Reference to the difference curve of Fig. 3 shows that if, for example, a
source of speech is 20 degrees to the right of the median plane the speec h
heard by the right ear is 3 db louder than that heard by the left ear.
A similar difference exists when the angle is 167 degrees. Presumably,
when the right ear hears speech 3 db louder than the left, the observer
localizes the sound as coming from a position 20 degrees or 167 degrees
to the right, depending upon the quality of the speech. If this be
assumed to be true, even though the difference is caused by the com-
bination of sounds of similar quality from several sources, it should be
possible to calculate the apparent angle.

PHYSICAL FACTORS

253

J -5
ffi

NESS DIFFERENCE IN DE

U> O IT O

V-

•^

Q -JO

-35

500 1000 5000

FREQUENCY IN CYCLES PER SECOND

20,000

Fig. 4 — Loudness difference produced in the right ear when a source of pure tone is
moved from the right to the left of an observer.

Loudness Theory of Localization
Upon this assumption the apparent angle of the source as a function
of the difference in decibels between the speech levels emitted by the
loud speakers of the 2- and 3-channel systems has been calculated.
Each loud speaker contributes an amount of direct sound loudness to
each ear, depending upon its distance from, and its angular position
with respect to, the observer. These contributions were combined on
a power basis to give a resultant loudness of direct sound at each ear,
from which the difference in loudness between the two ears was deter-
mined. The calculated results for the 2- and 3-channel systems are
shown by the solid lines in Fig. 5. The y axis shows the apparent
angle, positive angle being measured in a clockwise direction. The
X axis shows the difference in decibels between the speech levels from
the right and left loud speakers. The points are observed values
taken from Fig. 1. The observed apparent angles were obtained
directly from the average observer's location and the average apparent
positions shown in Fig. L The speech levels from each of the loud
speakers were calculated for each position on the pick-up stage. This
was done by assuming that the waves arriving at the microphone had
relative levels inversely proportional to the squares of the distances
traversed. By correcting for the angle of incidence and for the known
relative gains of the systems, the speech levels from the loud speakers
were obtained.

A comparison of the observed and calculated results seems to indi-
cate that the loudness difference at the two ears accounts for the greater
part of the apparent angle of the reproduced sounds. If this is true.

254

BELL SYSTEM TECHNICAL JOURNAL

the angular location of each position on the virtual stage results from a
particular loudness difference at the two ears produced by the speech
coming from the loud speakers. When three channels are used a definite

^ -20

2 CHANNELS

•^--^'

."""''^

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— — •

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Fig. 5 — Calculated and observed apparent angles for 2- and 3-channel reproduction.

set of loud speaker speech levels exists for each position on the pick-up
stage. To create these same sets of loud speaker speech levels with
the 3-microphone 3-loud speaker bridging arrangement already dis-
cussed, it would be necessary to change the bridging gains for each
position on the pick-up stage. Hence it could not be expected that
the arrangement as used (i.e., with fixed gains) would create a virtual
stage identical with that created by 3-channel reproduction. How-
ever, with proper technique, bridging arrangements on a given number
of channels can be made to give better reproduction than would be
obtained with the channels alone.

Experimental Verification of Theory
Considerations of loudness difference indicate that all caller positions
on the pick-up stage giving the same relative loud speaker outputs

PHYSICAL FACTORS

255

should be localized at the same virtual angle. The solid lines of Fig. 6
show a stage layout used to test this hypothesis with the 2-channel
system. All points on each line have a constant ratio of distances to

C-SCHANNELS
0°-3 CHANNELS

LEFT

CENTER
MICROPHONES

Fig. 6 — Pick-up stage contour lines of constant apparent angle.

the microphones. The resulting direct sound differences in pressure
expressed in decibels and the corresponding calculated apparent angles
are indicated beside the curves. The apparent angles were calculated
for an observing position on a line midway between the two loud speak-
ers but at a distance from them equal to the separation between them.
The microphones were turned face up at the height of the talker's
lips to eliminate quality changes caused by changing incidence angle.
It was found that a caller walking along one of these lines maintained
a fairly constant virtual angle. For caller positions far from the
microphones the observed angles were somewhat greater than those
computed. For highly reverberant conditions, the tendency was
toward greater calculated than observed angles. Reverberation also
decreased the accuracy of localization.

A change of relative channel gain caused a change in virtual angle
as would be expected from loudness difference considerations. For
instance, if the caller actually walked the left 3-db line, he seemed to
be on the 6-db line when the left channel gain was raised 3 db. Many
of the effects of moving about the pick-up stage could be duplicated
by volume control manipulation as the caller walked forward and
backward on the center path. With a bridged center microphone
substituted for the two side microphones similar effects were possible
and, in addition, the caller by speaking close to the microphone could
be brought to the front of the virtual stage.

256 BELL SYSTEM TECHNICAL JOURNAL

For observing positions near the center of the auditorium the
observed angles agreed reasonably well with calculations based only
upon loudness differences. As the observer moved to one side, how-
ever, the virtual source shifted more rapidly toward the nearer loud
speaker than was predicted by the computations. This was true of
reproduction in the auditorium, both empty and with damping simu-
lating an audience, and outdoors on the roof. Computations and
experiment also show a change in apparent angle as the observer
moves from front to rear, but its magnitude is smaller than the error
of an individual localization observation. Consequently, observers in
different parts of the auditorium localize given points on the pick-up
stage at different virtual angles.

Because the levels at the three microphones are not independent,
and because the desired contours depend upon the effects at the ears,
a 3-channel stage is not as simple to lay out as a 2-channel stage. For
a given observing position, however, a set of contour lines can be cal-
culated. The dashed lines at the right of Fig. 6 show four contours
thus calculated for the circuit condition of Fig. 1 and the observing
position previously mentioned. The addition of the center channel
reduces the virtual angle for any given position on the pick-up stage
by reducing the resultant loudness difference at the ears. Although
the 3-channel contours approach the 2-channel contours in shape at the
back of the stage, a given contour results in a greater virtual angle for
2- than for 3-channel reproduction.

Similar effects were obtained experimentally. As in 2-channel
reproduction, movements of the caller could be simulated by manipu-
lation of the channel gains. From an observing standpoint the 3-
channel system was found to have an important advantage over the
2-channel system in that the shift of the virtual position for side
observing positions was smaller.

Effects of Quality

If the quality from the various loud speakers differs, the quality of
sound is important to localization. When the 2-channel microphones
were so arranged that one picked up direct sound and reverberation
while the other picked up mostly reverberation, the virtual source was
localized exactly in the "direct" loud speaker until the power from
the "reverberant" loud speaker was from 8 to 10 db greater. In gen-
eral, localization tends toward the channel giving most natural or
"closeup" reproduction, and this effect can be used to aid the loud-
ness differences in producing angular localization.

PHYSICAL FACTORS 257

Principal Conclusions
The principal conclusions that have been drawn from these inves-
tigations may be summarized as follows:

1. Of the factors influencing angular localization, loudness difference
of direct sound seems to play the most important part; for certain
observing positions the effects can be predicted reasonably well from
computations. When large quality differences exist between the
loudspeaker outputs, the localization tends toward the more natural
source. Reverberation appears to be of minor importance unless
excessive.

2. Depth localization was found to vary with changes in loudness,
the ratio of direct to reverberant sound, or both, and in a manner not
found subject to computational treatment. The actual ratio of direct
to reverberant sound, and the change in the ratio, both appeared to
play a part in an observer's judgment of stage depth.

3. Observers in various parts of the auditorium localize a given
source at different virtual positions, as is predicted by loudness com-
putations. The virtual source shifts to the side of the stage as the
observer moves toward the side of the auditorium. Although quan-
titative data have not been obtained, qualitative data on these effects
indicate that the observed shift is considerably greater than that
computed. Moving backward and forward in the auditorium appears
to have only a small effect on the virtual position.

4. Because of these physical factors controlling auditory perspective,
point-for-point correlation between pick-up stage and virtual stage
positions is not obtained for 2- and 3-channel systems. However,
with stage shapes based upon the ideas of Fig. 7, and with suitable
use of quality and reverberation, good auditory perspective can be
produced. Manipulation of circuit conditions probably can be used
advantageously to heighten the illusions or to produce novel effects.

5. The 3-channel system proved definitely superior to the 2-channel
by eliminating the recession of the center-stage positions and in re-
ducing the differences in localization for various observing positions.
For musical reproduction, the center channel can be used for inde-
pendent control of soloist renditions. Although the bridged systems
did not duplicate the performance of the physical third channel, it is
believed that with suitably developed technique their use will improve
2-channel reproduction in many cases.

6. The application of acoustic perspective to orchestral reproduction
in large auditoriums gives more satisfactory performance than probably
would be suggested by the foregoing discussions. The instruments
near the front are localized by every one near their correct positions.

258 BELL SYSTEM TECHNICAL JOURNAL

In the ordinary orchestral arrangement, the rear instruments will be
displaced in the reproduction depending upon the listener's position,
but the important aspect is that every auditor hears differing sounds
from differing places on the stage and is not particularly critical of the
exact apparent positions of the sounds so long as he receives a spatial
impression. Consequently 2-channel reproduction of orchestral music
gives good satisfaction, and the difference between it and 3-channel
reproduction for music probably is less than for speech reproduction
or the reproduction of sounds from moving sources.

References

1. "Some Physical Factors Affecting the Illusion in Sound Motion Pictures," J. P.

Maxfield. Jour. Acous. Soc, July, 1931.

2. "Minimum Audible Sound Fields," L. J. Sivian and S. D. White. Jour. Acous.

Soc, April, 1933.

Loud Speakers and Microphones*

By E. C. WENTE and A. L. THURAS

In ordinary radio broadcast of symphony music, the effort is to create
the effect of taking the Hstener to the scene of the program, whereas in
reproducing such music in a large hall before a large gathering the effect
required is that of transporting the distant orchestra to the listeners. Lack-
ing the visual diversion of watching the orchestra play, such an audience
centers its interest more acutely in the music itself, thus requiring a high
degree of perfection in the reproducing apparatus both as to quality and
as to the illusion of localization of the various instruments. Principles of
design of the loud speakers and microphones used in the Philadelphia-
Washington experiment are treated at length in this paper.

AS EARLY as 1881 a large scale musical performance was repro-
■ duced by telephone instruments at the Paris Electrical Exhibition.
Microphones were placed on the stage of the Grand Opera and con-
nected by wires to head receivers at the exposition. It is interesting
to note that separate channels were provided for each ear so as to give
to the music perceived by the listener the "character of relief and
localization." With head receivers it is necessary to generate enough
sound of audible intensity to fill only a volume of space enclosed
between the head receiver and the ear. As no amplifiers were avail-
able, the production of enough sound to fill a large auditorium would
have been entirely outside the range of possibilities. With the advent
of telephone amplifiers, microphone efficiency could be sacrificed to
the interest of good quality where, as in the reproduction of music,
this was of primary interest. When amplifiers of greater output power
capacity were developed, loud speakers were introduced to convert a
large part of the electrical power into sound so that it could be heard
by an audience in a large auditorium. Improvements have been made
in both microphones and loud speakers, resulting in very acceptable
quality of reproduction of speech and music; as is found, for instance,
in the better class of motion picture theaters.

In the reproduction, in a large hall, of the music of a symphony
orchestra the approach to perfection that is needed to satisfy the
habitual concert audience undoubtedly is closer than that demanded
for any other type of musical performance. The interest of the listener
here lies solely in the music. The reproduction therefore should be

* Third paper in the Symposium on Wire Transmission of Symphonic Music
and Its Reproduction in Auditory Perspective. Presented at Winter Convention of
A. I. E. E., New York City, Jan. 23-26, 1934. Published in Electrical Engineering,
January, 1934.

259

260 BELL SYSTEM TECHNICAL JOURNAL

such as to give to a lover of symphonic music esthetic satisfaction at
least as great as that which would be given by the orchestra itself
playing in the same hall. This is more than a problem of instrument
design, but this paper will be restricted to a discussion of the require-
ments that must be met by the loud speakers and microphones, and
to a description of the principles of design of the instruments used in
the transmission of the music of the Philadelphia Orchestra from
Philadelphia to Constitution Hall in Washington. Some of the
requirements are found in the results of measurements that have been
made on the volume and frequency ranges of the music produced by
the orchestra.

General Considerations

The acoustic powers delivered by the several instruments of a
symphony orchestra, as well as by the orchestra as a whole, have been
investigated by Sivian, Dunn, and White. Figure 1 was drawn on
the basis of the values published by them.^ The ordinates of the
horizontal lines give the values of the peak powers within the octaves
indicated by the positions of the lines. For a more exact interpretation
of these values the reader is referred to the original paper, but the
chart here given will serve to indicate the power that a loud speaker
must be capable of delivering in the various frequency regions, if the
reproduced music is to be as loud as that given by the orchestra itself.
However, it was the plan in the Philadelphia-Washington experiment
to reproduce the orchestra, when desired, at a level 8 or 10 db higher,
so that with three channels each loud speaking system had to be able
to deliver two or three times the powers indicated in Fig. 1. Sivian,
Dunn, and White also found that for the whole frequency band the
peak powers in some cases reached values as high as 65 watts. In
order to go 8 db above this value, each channel would have to be capa-
ble of delivering in the neighborhood of 135 watts.

The chart (Fig. 1) shows that the orchestra delivers sound of com-
parable intensity throughout practically the whole audible range.
Although it is conceivable that the ear would not be capable of
detecting a change in quality if some of the higher or lower frequencies
were suppressed, measurements published by W. B. Snow ^ show that
for any change in quality in any of the instruments to be undetectable
the frequency band should extend from about 40 to about 13,000 c.p.s.
The necessary frequency ranges that must be transmitted to obviate
noticeable change in quality for the different orchestral instruments
are indicated in the chart of Fig. 2, which is taken from the paper by
Snow.

LOUD SPEAKERS AND MICROPHONES

261

t- 8

z
o

62.5 125 250 500 1000 2000 4000 11,500

FREQUENCY IN CYCLES PER SECOND

Fig. 1 — Peak powers delivered by an orchestra within various frequency regions.

TYMPANI

BASS DRUM

SNARE DRUM

14 INCH CYMBALS

BASS VIOL

CELLO

PIANO

BASS TUBA

TROMBONE

FRENCH HORN

TRUMPET

BASS SAXOPHONE

BASSOO^J

BASS CLARINET

CLARINET

SOPRANO SAXOPHONE-
OBOE

FLUTE

PICCOLO

100 500 1000 5000 10,000 20,000

FREQUENCY IN CYCLES PER SECOND

r"'S- 2 — Frequency transmission range required to produce no noticeable distortion
for orchestral instruments.

262

BELL SYSTEM TECHNICAL JOURNAL

Thus far only the sound generated by the orchestra itself has been
considered. However, it is well known that the esthetic value of
orchestral music in a concert hall is dependent to a very great extent
upon the acoustic properties of the hall. At first thought one might
be inclined to leave this out of account in considering the reproduction
by a loud speaking system, as one should normally choose a hall known
to have satisfactory acoustics for an actual orchestra. There would
be no further problem in this if the orchestral instruments and the
loud speaker radiated the sound uniformly in all directions, but some
of the important instruments are quite directive; i.e., they radiate
much the greater portion of their sound through a relatively small
angle. As an example, a polar diagram giving the relative intensities
of the sound radiated in various directions by the violin is given in
Fig. 3, which is taken from a paper published by Backhaus.^ The

7>>>v

t /\ fv^A K

/ /^^\/ /\ ^

<c / \ V\ \ \ \

/ / /^^^ / ^v 0

^^/\ V\\ \\\

/ / / >( y\\

r / \ /\ \ \ 3r-^'M

\ 1 ~^^ / / ^\ y^ \ '^

,o/ Nr • \\ y<c\ \

\ / /^^^^ / ^x^ \y^

40/C /\ \V\ \ \

^m\\

Fig. 3 — Variation of intensity with direction of the sound radiated by a violin

(660 c.p.s.).

directional characteristics of some of the instruments is one of the
chief reasons why the music from an orchestra does not sound the
same in all parts of a concert hall. The music which we hear comes to
us in part directly and in. part indirectly; i.e., after one or more
reflections from the walls. Both contribute to the esthetic value of
the music. The ratio of the direct to the indirect sound, which has
been designated by Hughes '' as the acoustic ratio, is to a first approx-
imation inversely proportional to the product of the reverberation
time and the angle through which the sound is radiated.^ For a
steady tone by far the greater part of the intensity at a given point in
a hall remote from the source is attributable to the indirect sound.
However, inasmuch as many of the tones of a musical .selection are

LOUD SPEAKERS AND MICROPHONES 263

of short duration, the direct sound is of great importance; it is this
sound alone which enables us to localize the source. So far as this
ratio is concerned, a decrease in the radiating angle of a loud speaker
is equivalent to a reduction in the reverberation time of the hall. The
efifect on the music, however, is not entirely equivalent, for the rate
of decay of sound in the room is unaltered by a change in directivity
of the source, as this depends only on the reverberation time.

As already pointed out, some of the instruments of the orchestra
are quite directive and others are nondirectional. In general, it may
be said that the instruments of lower register are less directive than
those of higher register. To have each instrument as reproduced by
the loud speaker sound just as the instrument itself would sound in
the same hall, the loud speaker would have to reproduce the music
from each instrument with a directivity corresponding to that of the
instrument itself. This manifestly is impossible. The best that can
be hoped for is a compromise. Let the loud speaking system be
designed so that it is nondirective for the lower frequencies, and at the
higher frequencies it will radiate the sound through a larger angle than
the most directive of the instruments and through a smaller angle than
the least directive. Although this compromise means that the
individual instruments will not sound exactly like the originals, it
carries with it one advantage: At all the seats in the hall included in
the radiating angle and at a given distance from the loud speaker the
music may be heard to equal advantage, whereas with the orchestra
itself the most desirable seats comprise only a certain portion of the
hall. The optimum radiating angle is largely a matter of judgment;
if it is too small the music will lack the spatial quality experienced at
indoor concerts; if it is too large there will be a loss in definition.

There is another respect in which the directivity of the source can
greatly affect the tone quality. Most loud speakers radiate tones of
low frequency through a relatively large angle, but as the frequency
is increased this angle becomes smaller and smaller. Under this
condition the relation between the intensities of the high and low
frequency tones as received directly will be different for almost all
parts of the hall. Hence, even with equalization by electrical net-
works, the reproduction at best can be good only at a few places in
the hall. Therefore, the sound radiated not only should be contained
within a certain solid angle, but the radiation throughout this angle
should be uniform at all frequencies.

The Loud Speaker
At present two kinds of loud speakers are in wide commercial use,
the direct radiating and the horn types. Each has its merits, but the

264 BELL SYSTEM TECHNICAL JOURNAL

latter was used in the Philadelphia-Washington experiment because
it appears to have definite advantages where such large amounts of
power are to be radiated. The horn type can be given the desired
directive properties more readily, and higher values of efficiency
throughout, a wide frequency range are more easily realized. In
consideration of the large power requirements, high efficiency is of
special importance because it will keep to the lowest possible value the
power capacity requirements of the amplifiers and because, with the
heating proportional to one minus the efficiency, the danger of burning
out the receiving units is reduced.

For efficiently radiating frequencies as low as 40 c.p.s., a horn of large
dimensions is required. In order that the apparatus may not become
too unwieldly the folded type of horn is preferable, but a large folded
horn transmits high frequency tones very inefficiently. As actually
used, therefore, the loud speaker was constructed in two units: one for
the lower and the other for the higher frequencies, an electrical network
being used to divide the current into two frequency bands, the point
of division being about 300 c.p.s.

The Low Frequency Horn
When moderate amounts of power are transmitted through a horn
the sound waves will suffer very little distortion, but when the power
per unit area becomes large, second-order effects, usually neglected in
considering waves of small amplitude, must be taken into account.
The transmission of waves of large amplitude through an exponential
horn has been investigated theoretically by M. Y. Rocard.^ His
investigation shows that if W watts are transmitted through the
throat of an exponential horn a second harmonic of intensity RW will
be generated, where R is given by the relation

(7 + \)T X WW

2pc\fM • ^^

in which/ is the frequency of the fundamental, /o the cut-off frequency
of the horn, c the velocity of sound, p the density of air, and A the
area of the throat of the horn, all expressed in c.g.s. units. It may
be noted that the intensity of the harmonic increases with the ratio
of the frequency to the cut-off frequency of the horn; this is another
argument against attempting to cover too wide a range of frequencies
with a single horn. In Fig. 1 it is shown that in the region of 200
c.p.s. the orchestra gives peak powers of about 10 watts. If, therefore,
30 watts be set as the limit of power that the horn is to deliver at 200
c.p.s., 32 c.p.s. as the cut-off frequency of the horn, and 30 db below

LOUD SPEAKERS AND MICROPHONES

265

the fundamental be assumed as the Umit of tolerance of a second
harmonic, from equation (1) a throat diameter of about 8 inches is
determined.*

If the radiation resistance at the throat of a horn is not to vary-
appreciably with frequency, the mouth opening must be a substantial
fraction of a wave-length. This condition calls for an unusually
large horn if frequencies down to 40 c.p.s. and below are to be trans-
mitted. However, the effect of variations in radiation resistance on
sound output can be kept down to a relatively small value if the
receiving unit is properly designed. This will be explained in the

2.8

2.4
2.2
2.0
1.8
1.6
1.4

1.0
0.8
0.6
0.4

0.2
0

-0.2

r
1 k

s I

— 1

\ /

I s

i\,

- /
/

\
\

1 i
1 /

^__

^ — h
1 1
1 /

1 /

\
\

/
/

\
\

— /-

/
/

—

\ —

1
-/~~

1 /
1 /

1 /

\ 1

1 /

0 40 80 120 160 200 240 280

FREQUENCY IN CYCLES PER SECOND

Fig. 4— Radiation resistance and reactance of low frequency horn.

next section. The low frequency horn used in these reproductions
has a mouth opening of about 25 square feet. As computed from well-
known formulas ^ for the exponential horn the impedance of this horn

* Since the original publication of this paper, experimental data have been ob-
tained which indicate a second harmonic genereition in horns 6 or more db below the
value shown by Rocard's equation.'

266 BELL SYSTEM TECHNICAL JOURNAL

with a throat diameter of 8 inches is shown in Fig. 4. These curves were
computed under the assumption that the mouth of the horn is sur-
rounded by a plane baffle of infinite extent, a condition closely approx-
imated if the horn rests on a stage floor.

Low Frequency Receiving Unit
When a moving coil receiving unit, coupled to a horn, is connected
to an amplifier having an output resistance equal to w — 1 times
the damped resistance R of the driving coil, it can easily be shown that
the sound power output is

P = -F J„.„ ■ . ./„-,. watts, (2)

^^ + ^

+ Ixa + T'xJ

where E is the open circuit voltage of the amplifier, L the length of
wire in the receiver coil, T the ratio of the area of the diaphragm to
the throat area of the horn, r + jx the throat impedance of the horn,
and Xd the mechanical reactance of the diaphragm and coil, the
mechanical resistance of which is assumed to be negligibly small.
From Fig. 4 it may be seen that the mean value of x increases as the
frequency decreases to a value below 40 c.p.s., and that x is smaller
than r except at the very lowest frequencies. If, therefore, the stiffness
of the diaphragm be adjusted so that Xd is equal to T^ times the mean
value of X at 40 c.p.s., the second term in the denominator may be
neglected without much error because it will have but little effect
upon the sound output except at the higher frequencies, where the
mass reactance of the coil and diaphragm may have to be taken into
account.

If minimum variations in sound output are desired for variations
in r,

B^mo-^ _ ...

where ro is equal to the geometric mean value of r, which is approx-
imately equal to Ape.

If a is the ratio of the resistance at any frequency to the mean
value, and if the second term in the denominator is neglected, equation
(2) becomes

P=^ ^. (4)

In Fig. 4 it is shown that above 35 c.p.s. a has extreme values of 2.75
and 0.36, at which points there will be minimum values in P, but these

LOUD SPEAKERS AND MICROPHONES 267

minimum values will not lie more than 1 db below the maximum

values. Hence, if the receiver satisfies the condition of equation (3),

the extreme variations in the sound output will not exceed 1 db,

although the horn resistance varies by a factor of 7.5. Also it may

be stated here that when the condition of equation (3) is satisfied the

horn is terminated at the throat end by a resistance equal to the surge

resistance of the horn. Thus equation (3) establishes a condition of

minimum values in the transient oscillations of the horn.

B'^U X 10~^

The mean motional impedance of the loud speaker is tp^ ,

-TVo

which, from equation (3), is equal to nR. The condition of equation

(3) therefore specifies that the efiiciency of the loud speaker shall be

— ; — 7 • The maximum power that an amplifier can deliver without
n -\- \

introducing harmonics exceeding a specified value is a function of the

impedance into which it operates. Therefore, to obtain the maximum

acoustic power for a specified harmonic content, the load impedance

should have the value for which the product of the loud speaker

efficiency and the power capacity of the amplifier has a maximum

value. This optimum value of load impedance for the amplifier and

loud speaker used in the Philadelphia-Washington experiments was

found to be about 2.25 times the output impedance of the amplifier;

the corresponding value of n then is 2.6 and the required efficiency

72 per cent. For best operating condition a definite value of receiver

efficiency thus is specified.

The receiver may be made to satisfy the foregoing conditions
regardless of the value of T, the ratio of diaphragm area to throat area.
The area of the diaphragm has, however, a definite relation to the
maximum power that the receiver can deliver at the low frequencies.
The peak power delivered by the receiver is equal to T^aro^^u^ X 10~^
peak watts where | is the maximum amplitude of motion of the
diaphragm. Figure 1 shows that in the region lying between 40 and
60 c.p.s., peak powers reach a value of from 1 to 2 watts. However,
the low frequency tones of an orchestra are undesirably weak and
may advantageously be reproduced at a relatively higher level.
Therefore it was decided to construct the loud speaker to be able to
deliver 25 watts in this region.

As the coil moves out of its normal position in the air gap, the force
factor varies. Harmonics thus will be generated, the intensities of
which increase with increasing amplitude. A limit to the maximum
value of the amplitude ^ thus is set by the harmonic distortion that
one is willing to tolerate. In this receiver the maximum value of ^

268 BELL SYSTEM TECHNICAL JOURNAL

was taken equal to 0.060 in. Figure 4 shows that aco^ has a minimum
value at about 50 cycles, where a is equal to about 0.4. These values
give a ratio of 4.5 for T.

Inasmuch as i? = • , where a is the resistivity of the wire used

for the coil and v the volume of the coil, from equation (3) is obtained

Bh = naTholO\ (5)

The first member gives the total magnetic energy that must be set
up in the region occupied by the driving coil. This value is fixed by
the fact that all factors in the second member are specified. The
same performance is obtained with a small coil and high flux density
as with a large coil and low flux density, provided B'^v is held fixed,
but the coil in any case should not be made so small that it will be
incapable of radiating the heat generated within it without danger of
overheating, nor so large that the mass reactance of the coil will
reduce the efficiency at the higher frequencies.

This receiver unit, when constructed according to the above prin-
ciples and when connected to an amplifier and a horn in the specified
manner, should be capable of delivering power 3 or 4 times that de-
livered by the orchestra in the frequency region lying between 35 and
400 c.p.s., with an efficiency of about 70 per cent, and with a variation
in sound output for a given input power to the amplifier of not more
than 1 db throughout this range.

The High Frequency Horn

It is well known that a tapered horn of the ordinary type has a
directivity which varies with frequency. Sound of low frequency is
projected through a relatively large angle. As the frequency is
increased this angle decreases progressively until, at frequencies for
which the wave-length is small compared with the diameter of the
mouth opening, the sound beam is confined to a very narrow angle
about the axis of the horn.

If we had a spherical source of sound (i.e., a source consisting of a
sphere, the surface of which has a radial vibratory motion equal in
phase and amplitude at every point of the surface), sound would be
radiated uniformly outward in all directions; or, if we had only a
portion of a spherical surface over which the motion is radial and
uniform, uniform sound radiation still would prevail throughout the
solid angle subtended at the center of curvature by this portion of the
sphere, provided its dimensions were large compared with the wave-
length. Throughout this region the sound would appear to originate

LOUD SPEAKERS AND MICROPHONES

269

at the center of curvature. Hence, for the ideal distribution of a
spherical source within a region to be defined by a certain solid angle,
it is necessary and sufficient that the radial motion be the same in
amplitude and phase over the part of a spherical surface intercepted
by the angle and having its center of curvature at the vertex and
located at a sufficient distance from the vertex to make its dimensions
large compared with the wave-length. If, further, these conditions
are satisfied for this surface at all frequencies, all points lying within
the solid angle will receive sound of the same wave form. A horn was
designed to meet these requirements for the high frequency band.

Fig. 5 — Special loud speaker developed for auditory perspective experiment.

270 BELL SYSTEM TECHNICAL JOURNAL

The horn, shown in the upper part of Fig. 5, comprises several
separate channels, each of which has substantially an exponential
taper. Toward the narrow ends these channels are brought together
with their axes parallel, and are terminated into a single tapered tube
which at its other end connects to the receiver unit. Sound from the
latter is transmitted along the single tube as a plane wave and is
divided equally among the several channels. If the channels have
the same taper, the speed of propagation of sound in them is the same.
The large ends are so proportioned and placed that the particle motion
of the air will be in phase and equal over the mouth of the horn. This
design gives a true spherical wave front at the mouth of the horn at
all frequencies for which the transverse dimensions of the mouth
opening are a large fraction of a wave-length.

As the frequency is increased, the ratio of wave-length to transverse
width of the channels becomes less, and the sound will be confined
more and more to the immediate neighborhood of the axis of each
channel. The sound then will not be distributed uniformly over the
mouth opening of the horn, but each channel will act as an independent
horn. To have a true shperical wave front up to the highest fre-
quencies, the horn would have to be divided into a sufficient number
of channels to make the transverse dimension of each channel small
compared with the wave-length up to the highest frequencies. If it
is desired to transmit up to 15,000 c.p.s., it is not very practical to
subdivide the horn to that extent. Both the cost of construction and
the losses in the horn would be high if designed to transmit also
frequencies as low as 200 c.p.s., as is the case under consideration.
However, it is not important that at very high frequencies a spherical
wave front be established over the whole mouth of the horn. For
this frequency region it is perfectly satisfactory to have each channel
act as an independent horn, provided that the construction of the
horn is such that the direction of the sound waves coming from the
channels is normal to the spherical wave front.

The angle through which sound is projected by this horn is about
60 degrees, both in the vertical and in the horizontal direction. For
reproducing the orchestra two of these horns, each with a receiving unit,
were used. They were arranged so that a horizontal angle of 120 degrees
and a vertical angle of 60 degrees were covered. These angular exten-
sions were sufficient to cover most of the seats in the hall with the loud
speaker on the stage. The vertical angle determines to a large extent
the ratio of the direct to the indirect sound transmitted to the audience.
The vertical angle of 60 degrees was chosen purely on the basis of judg-
ment as to what this ratio should be for the most pleasing results.

LOUD SPEAKERS AND MICROPHONES 271

The High Frequency Receiving Unit

In the design of the low frequency receiver one of the main objectives
was to reduce to a minimum the variations in sound transmission
resulting from variations in the throat impedance of the horn. How-
ever, the high frequency horn readily can be made of a size such that
the throat resistance has relatively small variations within the trans-
mitting region. On the other hand, whereas the diameter of the
diaphragm of the low frequency unit is only a small fraction of the
wave-length, that of the high frequency unit must be several wave-
lengths at the higher frequencies in order to be capable of generating
the desired amount of sound. Unless special provisions are made there
will be a loss in efficiency because of differences in phase of the sound
passing to the horn from various parts of the diaphragm. The high
frequency receiver therefore was constructed so that the sound gener-
ated by the diaphragm passes through several annular channels.
There are enough of these channels to make the distance from any
part of the diaphragm to the nearest channel a small fraction of a
wave-length. These channels are so proportional that the sound
waves coming through them have an amplitude and phase relation
such that a substantially plane wave is formed at the throat of the
horn.

In the appendix it is shown that, for the higher frequencies where
the impedance of the horn may be taken as equal to pc times the
throat area and for the type of structure adopted, the radiation
resistance is equal to

' ]

pcaV^
and the reactance

. pea „

kVi^T^ + kH^ cot2 kl

fiT V
kl cot kl + ( -J- ) kl tan kl

(6)

(7)

where a is the area of the throat of the horn, T the ratio of the area of
the diaphragm to the throat area, k =^ w/c, and the other designations
are those indicated in Fig. 11. At the lower frequencies the resistance
is Th and the reactance T^x, where r and x are, respectively, the
resistance and reactance of the throat of the horn.

Equation 6 shows that at a given frequency, other conditions
remaining the same, the radiation resistance will have a maximum
value when / is approximately equal to ir/2k = c/4/. In Fig. 6 the
resistances as computed from equation (6) are plotted as a function

272

BELL SYSTEM TECHNICAL JOURNAL

of frequency for several values of hjw. It is seen from these curves
that the resistance at the higher frequencies is determined very
largely by the relation of hjiv but is independent of it at the lower
frequencies, where it is equal to pcaT'^. At the lower frequencies
where the mechanical impedance of the diaphragm is negligible, the
efficiency, as was the case for the low frequency receiver, depends

O 1.0

h 21 —

-^—\

■-1

A —

iiimr^

\W'

d /2
b-24

d - 2

--\

■^

'""^

. —

~~"~~^

;;;; —

- -C^

^ \^

^<\

FREQUENCY X

Fig. 6 — Load iin]K'dancc of speaker (liapliragm.

LOUD SPEAKERS AND MICROPHONES

273

upon the value of Bh' where v Is the volume of the coil, but at the higher
frequencies the efficiency decreases with increasing mass of the coil.
It is advantageous, therefore, to keep v small and to make B as large
as is practically possible. Values were selected to give the receiver
an efficiency of 55 percent at the lower frequencies. For these con-
ditions the relative sound power output was computed by equation (2)
on the assumption that the receiver was connected to an amplifier
having an output impedance equal to 0.45 times that of the receiver

-20

A- EFFICIENCY

B- RELATIVE OUTPUT WHEN
CONNECTED TO AMPLIFIER

500

10,000 20,000

Fig. 7 — Relative computed sound output of high frequency receiver.

1000
FREQUENCY

2000 5000

IN CYCLES PER SECOND

at the lower frequencies. Figure 7 shows the values so obtained.
Corresponding values obtained experimentally when the receiver was
connected to the horn previously described are shown in Figs. 8 and 9,
where the sizes of the rooms in which the values were obtained were,
respectively, 5000 and 100,000 cubic feet. Both of these curves differ
considerably from the computed curve, particularly as regards loss
at high frequencies. The curve of Fig. 8 shows less, and that of Fig. 9
more, loss at high frequencies. The computed curve, however, refers
to the total sound output, whereas the measured curves give average
values of sound intensity in a certain part of the room, values depend-
ent upon the acoustic characteristics of the room.

The number of high frequency receivers that must be used for each
transmitting channel is governed largely by the amount of power that
the system is to deliver before harmonics of an objectionable intensity

274

BELL SYSTEM TECHNICAL JOURNAL

a -15

/^.

, ^.XVnn^nf^lrJlJlKAp/lrV-rU.ii'C W^l/lf

f^!"

X 1'

1/ U '^K ' '

200

500 1000 2000

FREQUENCY IN CYCLES PER SECOND

10,000

Fig. 8 — Output-frequency characteristic of high frequency receiver as measured in

a small room.

-10

-20

-30
40

— -

■^

Fig.

100 500 1000

FREQUENCY IN CYCLES PER SECOND

5000 10,000 20,000

9 — Output-frequency characteristic of combined low and high frequency re-
ceivers as measured in a large room.

9
45". 1

3°

x„.

—

\90°

100 500 1000

FREQUENCY IN CYCLES PER SECOND

5000 10.000 20.000

Fig. 10 — Output-frequency characteristic of moving coil microphone.

LOUD SPEAKERS AND MICROPHONES 275

are introduced. The generation of harmonics in a horn when trans-
mitting waves of large amplitude already has been discussed. Let it
suffice here to say that, for a given percentage harmonic distortion,
the power that can be transmitted through the horn is proportional
to the area of the throat and inversely proportional to the square of
the ratio of the frequency to the cut-off frequency.

Inasmuch as the moving coil microphones used for" the transmission
of music in acoustic perspective have been described previously ^ they
will not be discussed here at length. Their frequency response char-
acteristic as measured in an open sound field for several different
angles of incidence of the sound wave on the diaphragm are shown in
Fig. 10 where it is seen that the response at the higher frequencies
becomes less as the angle of incidence is increased. In general, this
is not a desirable property, but with the instruments as used in this
experiment the sound observed as coming from each loud speaker is
mainly that which is picked up directly in front of each microphone;
sound waves incident at a large angle do not contribute much.

At certain times the sound delivered by the orchestra is of very low
intensity. Therefore it is important that the microphones have a
sensitivity as great as possible, so that the resistance and amplifier
noises may readily be kept down to a relatively low value. At 1,000
c.p.s. these microphones, without an amplifier, will deliver to a trans-
mission line 0.05 microwatt when actuated by a sound wave having
an intensity of 1 microwatt per square centimeter. This sensitivity is
believed to be greater than that of microphones of other types hav-
ing comparable frequency response characteristics, with the possible
exception of the carbon microphone.

APPENDIX

Load Impedance of a Diaphragm Near a Parallel Wall with

Slot Openings

First assume a diaphragm and a parallel wall of infinite extent
separated by a distance h, and that the wall is slotted by a series of
equally spaced openings as shown in Fig. 11. From symmetry it is
known that when the diaphragm vibrates there will be no flow per-
pendicular to the plane of the paper or across the planes indicated by
the dotted lines. Therefore only one portion of unit width, such as
abcdef need be considered. Let the x and y reference axes be located
as shown. If the general field equation

d~(p d-(p
ox^ dy

. + 7^ + ^V = 0 (8)

276

BELL SYSTEM TECHNICAL JOURNAL

is applied when the diaphragm has a normal velocity equal ^c'"' the
following boundary conditions are obtained:

When X = 0, dcp/dx = — ^,

X = h, d(p/dx = 0,
y = 0, d^ldy = 0,

and when y = I, the pressure is equal to the product of acoustic
impedance and volume velocity or

21-

*-2w

i f

--2W

*-2w

Fig. 11 — Schematic diagram of diaphragm and parallel slotted wall of infinite length.

^r(^) jv^^rv-^^ dx

h Jo V dt I y=i w J^ \ dy I y^i

where <p is the velocity potential, k = oojc, and c is the velocity of
sound.

The appropriate solution of equation (8) then is

cos ^v

kh ( cos kl -]- i— sin kl

cos ^(.v — //.)
sin kh

The average reacting force per unit area of the diaphragm is

ikpc

y- I (<p)x=o

Thus, for the impedance per unit area, which is equal to the force
divided by the velocity, is obtained

pd]

sin^ kl

cos2 kl-\- { -] sin2 kl

It'

-J

kh cos kh
sin kh

kl -

sin kl cos kl

h \2
COS' kl -\- [ — I sin' kl

kH'

Y^ r' + jx'.

LOUD SPEAKERS AND MICROPHONES

277

In all practical types of loud speakers kh cos kh/sln kh would be
very nearly equal to 1 ; then

pel

X = —

pd
h

+ COt2 kl
1

cotkl + [~ ] tan kl

If the total area of the diaphragm is A and that of the corresponding
channels a, then Aja = l/iu, approximately, and the total impedance
becomes

pcA^ 1

/ kh \ 2

( — j A^ -\- kT- cot2 kl

X = - J

pcA

1 -

kl cot kl -\- [ J - I kl tan kl

References

1. "Absolute Amplitudes and Spectra of Certain Musical Instruments and Orches-

tra," L. J. Sivian, H. K. Dunn, and S. D. White. Jour. Acous. Soc. Am.,
V. 2, Jan. 1931, p. 330.

2. "Audible Frequency Ranges of Music, Speech, and Noise," W. B. Snow. Jo74r.

Acous. Soc. Am., v. 3, July 1931, p. 155.

3. Backhaus, Zeits. f. Tech. Physik, v. 9, 491, 1928.

4. "Engineering Acoustics," L. E. C. Hughes, p. 47. Benn, London.

5. W. J. Albersheim and J. P. Maxfield, similar relations were presented in a paper

before the Acoustical Society in May 1932.

6. "Sur la Propagation des Ondes Sonores d'Amplitude Finie," M. Y. Rocard.

Comptes Rendus, Jan. 16, 1933, p. 161.

7. "Theory of Vibrating Systems and Sound," Crandall. P. 163 ff. D. Van Nos-

trand. New York.

8. "Moving Coil Telephone Receivers and Microphones," E. C. Wente and A. L.

Thuras. Jour. Acous. Soc. Am., v. 3, 1931, p. 44.

9. " E.xtraneous Frequencies Generated in Air Carrying Intense Sound Waves,"

A. L. Thuras, R. T. Jenkins and H. T. O'Neil. To be presented at mtg. of
Acous. Soc. Am., April 30-May 1, 1934.

Amplifiers*

By E. O. SCRIVEN

Appreciable care is required in the design of a system which must amplify
with great fidelity practically the whole range of audible frecjuencies and
be capable of delivering a high level while at the same time providing a
wide volume range. Some of the problems involved are discussed, particu-
larly as applying to the equipment used in the reproduction in Washington,
D. C, of the Philadelphia Symphony Orchestra playing in Philadelphia.

VACUUM tube amplifiers have been closely identified with the
extension of the channels of communication since, with com-
pletion of the initial transcontinental telephone line 20 years ago, they
first enabled New York to converse with San Francisco. There are
now thousands of audio frequency amplifiers in telephone circuits and
in sound picture theaters, public address systems, and other similar
services as well as in the millions of radio receiving sets.

Along with the extension of the field of usefulness of audio amplifiers
there has been continuing progress toward more faithful reproduction,
better transmitters, better receivers, and better amplifiers. Those
first telephone repeaters, although quite adequate for their immediate
purpose, transmitted a frequency band only a few octaves wide.
Very few radio sets even now cover a range above 3,000 c.p.s. without
distortion, and the most up-to-date sound picture installation rarely
can be depended upon for accurate reproduction of frequencies above
7000 or 8000 c.p.s. The requirements as to frequency range and
freedom from distortion for any particular service are, in the last
analysis, determined by public demand.

However, when one undertakes to reproduce an orchestra like the
Philadelphia Symphony and to reproduce it in such a manner as to
satisfy the critical ear of the director, or that of the devotee of sym-
phonic concerts, one has to provide something out of the ordinary in
audio amplifiers.

In his paper, which forms a part of this symposium. Dr. Fletcher
has pointed out that only the elimination of those frequencies below
40 c.p.s. and those above 15,000 c.p.s. produces no detectable differ-
ence in the reproduction of symphonic music. This, then, is the

* Fourth paper in the Symjiosium on Wire Transmission of Symphonic Music
and Its Reproduction in Auditory Perspective. Presented at Winter Convention of
A. I. E. E., New York City, Jan. 23-26, 1934. Published in Electrical Engineering,
January, 1934.

278

AMPLIFIERS 279

frequency spectrum that the ampHfier must be designed to handle.
Also, it is important that there shall be uniform amplification of all
parts of the frequency range and that no extraneous frequencies shall
be introduced.

Of importance commensurate with the distortionless amplification
of the complete frequency range of the orchestra is the provision of an
equivalent volume of sound. The amplifier must be capable of supply-
ing to the loud speakers without distortion an amount of energy that
will produce a sound volume at least equivalent to that produced b>'
the orchestra (the Philadelphia-Washington installation was designed
to produce about 10 times this amount). And equally important, the
amplifier must be so free from internal disturbances and from self-
induced electrical fluctuations that the softest music, the weakest input
to the microphone, can be reproduced without appreciable background
noise. According to Fletcher the ratio of the heaviest playing of a
large orchestra such as the Philadelphia Symphony Orchestra to the
softest music such as that of a violin is about 10,000,000 to 1, or 70 db.
Thus it is required that any noise be at least 75 db below the loudest
tones; that is, there must be at least a 75-db volume range.

The sources of noise may be divided into 2 groups. In the first
group are included the 60-cycle alternating current power supply,
vibration or jar of mechanically unstable vacuum tubes, contact and
thermoelectric potentials, and similar disturbances, which may be
reduced to practically any degree depending upon the lengths to which
one is willing to go to reduce them. In the second group are those
electronic irregularities intimately associated with the operation of
the vacuum tube and which depend somewhat upon the design, manu-
facture, and method of operation of the vacuum tube; and which,
when sufficiently amplified and fed into a loud speaker, may be heard
as noise. In general, the maximum volume range of an amplifier is
reached when all other disturbances are reduced to the level of this
tube noise.

It is evident, then, that under ordinary circumstances the limiting
volume range of an amplifier is a function of the amount of ampli-
fication following the first tube. In other words, the magnitude of the
signal voltage with respect to the noise voltage in the plate circuit of
the first tube in a multistage amplifier determines the limiting volume
range obtainable with that amplifier.

It will appear that in a sound reproduction system a highly efficient
microphone simplifies the amplifier volume range requirements, and
that loud speakers of high efficiency reduce the volume required from
the amplifier.

280 BELL SYSTEM TECHNICAL JOURNAL

Perhaps it is in order to inquire as to what makes an ampUfier free
from frequency distortion over a wide range. The answer might well
be: attention to impedance relations. A compact, efficient amplifier
requires several pieces of reactive apparatus such as transformers,
retardation coils, and capacitors. One must remember that an in-
ductance of one henry is equivalent to an impedance of 250 ohms at
40 c.p.s. but that it is nearly 100,000 ohms at 15,000 c.p.s. ; that the
grid circuit of the vacuum tube is not actually an open circuit even
though the grid is maintained negative with respect to the cathode,
but has a reactance which becomes important at high frequencies or
with large ratio input transformers. Many years of development in
this field have advanced the art to the point where transformers
transmitting extremely wide bands now can be designed. The com-
mercial production of such designs requires rigid inspection including
shop transmission measurements under the actual conditions of use.
The transformer must be designed for the particular type of vacuum
tube with which it is to be used. First, however, the tube must be
designed to permit its use under the proposed conditions and then it
must be manufactured to close limits, every tube of a type like every
other tube of that type.

This is, then, the general requirement for a wide frequency range
amplifier: (1) attention to impedance relations; (2) meticulous design
of each component for the particular job it has to do, and rigid inspec-
tion to insure that it does that job.

One might suppose that when the tube designer and the coil designer
each had done his part the job was done. Such is not the case. The
various pieces of apparatus have to be gathered together into a unit
(often a current supply set for supplying anode, cathode, and grid
potential is assembled with the amplifier) and out of this electrical and
physical association is apt to arise "feed-back" and "noise."

When there is coupling between two parts of the amplification circuit
which are at different potential or different phase there is feed-back.
Feed-back sometimes is employed designedly to modify an amplifier
characteristic, but, feed-back which may arise as a result of a particular
arrangement of apparatus or wiring ordinarily will cause more or less
severe frequency distortion. It may be induced due to stray electro-
magnetic or electrostatic fields, which must be eliminated by rear-
rangement of apparatus or by shielding; or it may be caused by
common circuit impedance, requiring circuit modifications. In
general, a low gain amplifier or one with limited frequency range
presents no feed-back problems, but a study of a high-gain wide-
range equipment usually is necessary in order to determine the best

AMPLIFIERS 281

arrangement. Often modifications of tentative circuit or apparatus
must be made to obtain satisfactory operation.

The provision of a volume range of some 75 db on an energy basis
became largely a matter of the suppression of a.-c. hum. The low
inherent electronic noise effect of the Western Electric No. 262A vac-
uum tube and the relatively high level from the microphones kept
electronic tube noise well in the background. Careful and in some
cases rather elaborate shielding of audio transformers and leads and
the segregation of the 60-cycle power equipment coupled with the
use of vacuum tubes having indirectly heated cathodes and specially
designed to have small stray fields prevented a.-c. hum trouble in the
early stages. However, the Western Electric No. 242A vacuum tubes
used in the push-pull final stage have filamentary cathodes, and when
such tubes have raw a.-c. filament supply, a very appreciable 120-cycle
component appears in the space current. Although theoretically in a
perfectly balanced push-pull amplifier this component would be
eliminated, in practice an exact balance cannot be obtained. As a
final step in noise elimination, advantage was taken of the fact that
each channel employed two amplifiers in parallel. Under such condi-
tions and with proper phasing of the power supply to the two ampli-
fiers the net a.-c. noise output of the two amplifiers in parallel will be
less than that of either one alone.

Having reduced feed-back and noise to tolerable values, it remains
to determine the operating conditions for maximuum output. The
vacuum tube is not strictly a linear device, but, when properly used,
the total harmonic content can be held to a low figure. For a high
quality system the total harmonics produced in the system should
not exceed one per cent of the fundamental. This requires that
impedance and potential relations in the vacuum tubes should be
adjusted to give approximately linear operation; and also that the
design of audio transformers, particularly those carrying considerable
levels, must be scrutinized carefully to insure that they operate over
an essentially linear portion of the magnetization curve of the core
material.

An instrument really essential to the design of high quality amplifiers
is a high sensitivity harmonic analyzer that is capable of quickly and
accurately resolving a complex wave into its simple components. By
this means the effect of variations in circuit relations can be evaluated
and the optimum condition for maximum distortionless power output
determined.

It may be desirable at this point to examine the make-up of the
audio amplification system used in the Philadelphia-Washington
experiments. It should be noted that the arrangement of equipment

282

BELL SYSTEM TECHNICAL JOURNAL

AMPLIFIERS 283

provided for simultaneous reproduction at both Philadelphia and
Washington. There were three complete and essentially equivalent
channels of equipment actually in use and a fourth complete channel
held in reserve as a spare.

Several stages of so-called voltage amplification were required pre-
liminary to the final or power stage. There is, of course, no essential
difference between a voltage amplifier and a power amplifier, the term
"voltage amplifier" being applied to those preliminary stages of an
amplification system the function of which is so to amplify the output
of the pick-up device as to supply adequate driving voltage to the
grids of the power stage. Theoretically, inasmuch as no energy is
absorbed in the ideal grid circuit, this voltage increase might be
supplied entirely by a high ratio input transformer. However, there
are practical difficulties to the design of such a single stage amplifier
and therefore multistage vacuum tube amplification is employed.

As a matter of convenience the voltage amplification for this system
was obtained through the use of several separate amplifier units in
tandem. This arrangement not only enabled the ready replacement
of any unit of the system in case of failure, but it also facilitated the
insertion of a pad, control potentiometer, or other network at any
desired point. Several of these devices were required, and of course
each introduced a loss. Thus the gross amplification of the system
used for reproduction at Philadelphia was approximately 160 db and
for Washington 240 db, although the actual difference in level between
microphone output and loud speaker input was but from 80 to 90 db.

The general scheme of the amplification system is shown in Fig. 1.
yli is a single-stage, single-tube Western Electric No. 80A amplifier
^lightly modified to meet the particular conditions of use; it has a
gain of 30 db, and employs a Western Electric No. 262A vacuum tube.
This tube has an equipotential cathode, the heater being operated on
10-volt 60-cycle alternating current and the anode being supplied from
rectified alternating current. ^2 is a 2-stage amplifier having a single
Western Electric No. 262A vacuum tube in the first stage and push-
pull Western Electric No. 272A tubes in the second stage. It has a
gain of 50 db. The cathodes of the tubes are energized with low-
voltage 60-cycle alternating current and the anodes with rectified
alternating current. A^, the final or power amplifier, is a single stage
amplifier employing two Western Electric No. 242A vacuum tubes in
parallel on each side of a push-pull circuit, thus having four tubes per
amplifier. Two of the Az amplifiers were used in parallel on each
channel, and were capable of supplying 60 watts each, or a total of
120 watts, to the loud speakers. These are r.m.s. values. The instan-
taneous peaks of power of course could equal twice this value, or 720

284

BELL SYSTEM TECHNICAL JOURNAL

watts, for the three channels. Ei and E2 are equalizers to compensate
for any amplitude distortion that would cause a listener to obtain a
different tone effect from the loud speakers than he would from the

Fig. 2^Ainplifying equipment used at Philadelphia. The taller racks are 8 ft. high
and contain A\ and A-, amplifiers, volume indicators, and various controls.

actual orchestra. These equalizers are loss networks and principally
equalize for the acoustic characteristic of the loud speakers in the
particular hall, but they are placed in a low energy part of the ampli-
fication circuit so as not to waste the energy of the final power stage.
In view of the inclusion of the equalizers in the'amplification system,
and particularly because of the fact that the amplification of the Az
amplifier deliberately was made to increase with frequency in order
to compensate in part for acoustic losses in the overall system, the
actual amplification-frequency curves of the amplifiers are of little
importance. The equalizers of the system are discussed in the paper
by Bedell and Kerney.

Transmission Lines*

By H. A. AFFEL, R. W. CHESNUT and R. H. MILLS

Describing methods whereby high quality sound reproduction in auditory
perspective can be accomplished over long distances, this discussion centers
largely upon a description of the exact technique employed in providing
communication transmission circuits for the Philadelphia-Washington dem-
onstration. Problems that might be involved in carrying out such trans-
mission on a more widespread scale also are touched upon.

MICROPHONES have been described that will pick up without
noticeable distortion all the sounds given forth by a symphony
orchestra. Loud speakers and amplifiers also have been described
that will accurately reproduce this highest quality music in its full
range of tone quality and volume. Therefore, the situation obviously
requires connecting transmission paths so perfect in their character-
istics that reproduction 100 or 200 miles away may not suffer in com-
parison with reproduction which may be only 100 or 200 feet from the
source of music.

There are several respects in which a long line circuit possibly may
distort the speech or music passed over it, unless considerable effort
is expended to overcome these tendencies. For example, there may
be frequency-amplitude distortion; i.e., all the notes and overtones
may not be transmitted with the proper relative volumes. Similarly
there may be phase or delay distortion, the different frequencies may
not arrive at the receiving end of the line circuit in the same time
relationships in which they originated. A line circuit is subject also
to possible inductive disturbances from other communication circuits
("crosstalk"), or from power or miscellaneous circuits which cause
"noise" at the receiving terminal. If the circuit contains amplifiers,
transformers, and inductances having magnetic cores, it is subject to
possible nonlinearity effects; i.e., the current at the receiving end of
the line may not follow exactly the amplitude variations of the current
applied to the transmitting end or, what is more important, spurious
intermodulation frequencies may be generated within the transmission
circuit and mar the purity of the musical tones. The problem of
reproduction in auditory perspective, using two or three paralleling

* Fifth paper In the Symposium on Wire Transuiission of Symphonic Music

and Its Reproduction in Auditory Perspective. Presented at Winter Convention of

A. I. E. E., New York City, Jan. 23-26, 1934. Published In Electrical Engineering,
January, 1934.

285

286 BELL SYSTEM TECHNICAL JOURNAL

channels, also adds the requirement that these channels must be sub-
stantially identical in their transmission characteristics.

With the exception of the last, all these aspects of the problem are,
of course, not peculiar to symphony music transmission. They exist
as part of the problem of satisfactorily transmitting any telephone
message. However, the requirements of this new high quality trans-
mission have set a new high standard of refinement, even as compared
with that required for ordinary radio chain broadcasting. For ex-
ample, ordinary telephone message transmission commonly is carried
out by circuits having a frequency range not exceeding 200 to 3000
cycles per second. Much present-day radio broadcasting involves a
transmission band only from about 100 to 5000 c.p.s. This new high
quality transmission, however, requires a range from approximately
40 to 15,000 c.p.s. Further, with reference to the required freedom
from interference, ordinary radio broadcasting seldom exceeds a volume
range greater than 30 decibels. The new high-quality system, how-
ever, requires a volume range of at least 65 db, which is more than
3,000,000 to 1 expressed as a power ratio.

In considering the specific problem of transmitting from Philadelphia
to Washington for the demonstration given on April 27, 1933, several
alternative methods of providing the required transmission paths
presented themselves. The arrangement chosen consisted in bridging
the distance between the two cities by means of carrier channels over
cable conductors. From the telephone toll ofhce in Philadelphia to the
toll ofifice in Washington, three carrier transmission paths were provided
in which the music frequencies were stepped up from their normal
position in the audible range to considerably higher frequencies. The
frequency range from 40 to 15,000 c.p.s. picked up by the microphones
was transmitted over line circuits in a range from 25,000 to 40,000
c.p.s. After being thus stepped up in frequency, the high frequency
currents were applied to three non-loaded pairs in an all-underground
cable which was equipped with repeaters at approximately 25-mile in-
tervals. At Washington, step-down or demodulation apparatus restored
the frequencies to their normal position in the spectrum.

For transmission between the auditorium in Philadelphia and the
toll office there, a distance of approximately three miles, and for trans-
mission in Washington between the telephone toll office and the
auditorium, about half this distance, normal frequency transmission
over small-gauge pairs in ordinary exchange cables was employed.

The use of the carrier method for the long distance transmission has
several advantages. In general, it permits multiplex operation; i.e.,
more than one message or program on the same pair of wires. As a

TRANSMISSION LINES 287

matter of expediency in this particular case this feature of operation
was not used, and three separate pairs were employed, one for each
channel. In the future the same technique undoubtedly would permit
two or possibly more of these extra-broad-band transmission paths to be
obtained on the same pair of conductors. The most important reason
for choosing the carrier method rather than transmission in the natural
audio-frequency range in this particular case was that, because all
other transmission circuits in the same cable were at a considerably
lower frequency and because the lead sheath of the cable acts efficiently
at the high frequencies to shield the pairs from induced disturbances
from the outside, it offered a special freedom from crosstalk and noise.
With these arrangements, which will be described in somewhat
greater detail in what follows, requirements of transmission were met
very satisfactorily and the reproduction of the symphony music in
Washington with the orchestra playing in Philadelphia suffered not
the least in comparison with the reproduction of the same program in
an auditorium in Philadelphia located but a few feet from the hall in
which the orchestra played. It is believed that, if necessary, by the
use of the same principles, line circuits may be set up and comparable
quality reproduction given throughout the country. However, as
will be evident from part of the discussion which follows, in some
respects the problem of meeting the requirements in transmission
between Philadelphia and Washington was not as difficult as might
be encountered in other localities. Hence even more complex arrange-
ments might be necessary if it were desired to establish such trans-
mission circuits to other points, and particularly for greater distances.

Line Circuits

There are several all-underground cables between Philadelphia and
Washington. As described in a paper ^ by Clark and Green given
before the A. I. E. E. in 1930, recently laid cables contain several
16-gauge conductors distributed throughout the cross section of the
cable for possible use as program circuits in chain broadcasting.
These pairs, however, ordinarily are loaded and equipped with re-
peaters at approximately 50-mile intervals so that they transmit a
frequency range up to about 8000 c.p.s.

In one of the cables several pairs of this type had not yet been
loaded, and these pairs were used for this newer transmission. Because
of the higher frequencies employed and the greater attenuation en-
countered, it was necessary to install repeaters at more frequent
intervals. As may be noted in Fig. 1, the normal cable layout between
Philadelphia and Washington includes two intermediate repeater

288

BELL SYSTEM TECHNICAL JOURNAL

Cl, S

TRANSMISSION LINES

289

stations, one at Elkton and one at Baltimore. Additional repeater
stations were established accordingly at in-between points — Holly Oak,
Abingdon, and Laurel. One of these repeater points, Holly Oak, was
established in a local telephone office. No such convenient housing
existed at the other two points, and it was necessary to establish new
repeater stations. These were small metal structures large enough
to house only the repeaters, their power supply, and testing equipment.
This apparatus was arranged to be normally non-attended, various
switching actions being remotely controlled from the nearest regular
repeater station.

50
10

-I

CD 40
O

o
z

to 30
<r)

^"'^

4 8 12 16 20 24 28 32 36 40

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 2 — Line attenuation characteristic of typical repeater section.

The line attenuation between repeater points is shown in Fig. 2.
It may be noted that the attenuation is approximately 50 db for the
highest carrier frequency involved. A diagram showing the variations
in power level as the carrier waves traverse the complete circuit is
shown in Fig. 3. Because of the variation in attenuation over the

13 20

m 0

S-20

-^0

O-60

% A

J d

J D

Fig. 3 — Transmission level diagram.

frequency range employed it was necessary, of course, to use equalizers
at the input of each repeater; i.e., networks having an attenuation
variation with frequency approximately the inverse of that of the line
circuit.

290 BELL SYSTEM TECHNICAL JOURNAL

Noise

In setting up these circuits various tests, including measurements of
noise currents picked up by the conductors to be employed, were made
prior to the actual installation of the apparatus. It was discovered
that on the cable circuits north of Baltimore these pairs were picking
up sufficient noise even at the higher frequencies to constitute a
possible limitation in the volume range that might be delivered. This
noise was generated chiefly as a by-product of relay and other similar
operations w'ithin the Baltimore office and was propagated over the
longitudinal circuits of various pairs in the cable from which, by in-
duction, it entered the special selected pairs. As a remedial measure,
longitudinally acting choke coils applied to all but the specially
selected pairs in the cable greatly reduced the noise. Shielding and
physical separation were employed in the Baltimore office to prevent
induction between the repeaters and the connection to the main
cable. If it is desired to use existing cables for carrier transmission,
particularly for such high grade transmission circuits, it seems likely
that filtering arrangements of this kind, or other precautions, generally
will be required.

Carrier Apparatus

The carrier system employed may be characterized briefly as single-
sideband carrier-suppressed, with perfectly synchronized carrier fre-
quencies of 40,000 c.p.s. Most present-day commercial telephone
carrier systems are of the single-sideband carrier-suppressed type.
Suppressing one sideband saves frequency space and suppressing the
carrier reduces the load on the line amplifiers or repeaters. Ordinarily
the exact synchronization of the carrier frequencies at the sending and
receiving ends is not required for message telephone service.

Obtaining a single sideband after modulation commonly is carried
out by providing band filters which transmit the desired sideband and
suppress the unwanted sideband. For the requirements of message
telephone transmission this does not impose severe requirements in
the design of filters because audio frequencies less than about 100-200
c.p.s. ordinarily are not transmitted, in which case, if the filter in
suppressing the unwanted sideband tends to cut off the lower fre-
quencies of the desired sideband, it is not important.

For the requirements of this new high quality system, however,
where it was desired to transmit all frequencies to at least as low as
40 c.p.s., the problem was considerably more difficult. Two alter-
natives presented themselves in the design of the required filters.
The first consisted in attempting to provide the required selectivity
in the filters themselves, perhaps supplementing the actions of indue-

TRANSMISSION LINES 291

tance coils and condensers (which normally make up such a filter
structure) by quartz crystals to provide the sharp selectivity required
on the sides of the band. The other alternative consisted in providing
a filter of moderate selectivity so that in the neighborhood of the
carrier frequency the unwanted sideband is not completely suppressed,
and in arranging that the resultant reproduced music at the receiving
terminal is obtained by the proper coordination of the desired and
the vestige of the unwanted sideband. The "vestigial" sideband
method was decided upon. Although this does not require filters
having particularly sharp selectivity on the sides of the band, it does,
however, impose more severe requirements upon the control of the
phase characteristics of the filters in the neighborhood of the carrier
frequency. It makes it necessary also to have the carrier frequencies
at the sending and receiving ends not only synchronized, but phase
controlled as described later.

For the modulating elements in the system at both the sending and
receiving terminals, copper oxide rectifying disks were chosen. These
elements can be made very simple. In stability, with respect to
transmission loss and the ability to suppress the unwanted carrier
frequency by balanced circuits, this arrangement is superior to the
usual vacuum tube circuits.

In Fig. 4 is shown schematically the arrangements of the carrier
circuit at the transmitting and receiving ends. At the transmitting
terminal the circuit from the microphones is led first through low- and
high-pass filters to limit the bands to the desired width; i.e., 40 c.p.s.
to 15,000 c.p.s. The 40-cycle limiting filter was included because
tests had demonstrated that lower frequencies are not required for
the satisfactory transmission of music of symphony character, and
because it was feared that occasional high energy pulses of subaudible
frequency might cause overloading. When these 40-cycle filters are
omitted, as was done in tests, the carrier channels are capable of trans-
mitting frequencies down to and including zero frequency, a charac-
teristic which could not possibly be obtained in a single sideband
system by other than such a vestigial sideband technique.

As may be noted further in Fig. 4, carrier current is supplied to the
rectifying disks of the modulator along with the incoming music fre-
quencies. The balancing connection of the four rectifying disks mak-
ing up the modulator is arranged to suppress the carrier frequency, the
final degree of suppression being adjusted by means of the variable
condenser and resistance shown, which were included to make up for
slight dissimilarities in the characteristics of the individual copper
disks. A ver\' high degree of carrier suppression can be achieved by

292

BELL SYSTEM TECHNICAL JOURNAL

TRANSMISSION LINES 293

this means. No difficulty was experienced in maintaining a ratio of
at least 60 db between the carrier voltage applied to the unit and the
residual carrier current not completely balanced out. Over short
periods an even higher degree of balance can be readily obtained.

There is a certain amount of electrical noise generated in the recti-
fying disks over and above that caused by thermal agitation ^' ^
effects. The amount of this noise compared with the maximum per-
missible modulation output determines the volume range possibilities
of a modulator of this type. Measurements indicated that this range
was approximately 90 db, which obviously was more than sufficient to
meet the requirements desired.

The circuit includes a relay and a meter through both of which
flows the d.-c. component produced by the rectification of the carrier
frequency. These supplementary units give a check on the magnitude
of the carrier supply and afford an alarm in case of failure. From the
modulator unit the circuit is connected to the band filter which trans-
mits only the lower sideband lying between approximately 25,000 and
40,000 c.p.s. and the vestige of the upper sideband. From the band
filter the currents are led to an amplifier and thence to the line circuit
leading to the farther terminal.

It may be noted that at the transmitting terminal the 40,000-cycle
carrier current is derived from a 20,000-cycle oscillator by passing its
output through a series of copper oxide rectifiers connected to form a
frequency doubler. Part of the originally generated 20,000 cycles also
is connected to the input of the transmitting amplifier and sent over
the line to be used in producing the 40,000-cycle carrier supply for
demodulation.

At the receiving terminal a similar modulation or demodulation
process occurs through the use of copper oxide disk circuits. A relay
and meter also are included in the circuit to check the carrier supply,
in this case providing also a check or pilot of the transmission over the
long line circuit. The 20,000-cycle synchronizing current is selected
at the receiving terminal, amplified and applied to a frequency doubler,
and thence applied to the demodulator circuit. The input of this
carrier supply circuit includes also a phase adjusting variable con-
denser arrangement so that the phase of the carrier supplied to the
demodulator may be adjusted properly in relation to that of the
carrier supplied to the modulator at the sending end. An interesting
feature of the receiving terminal carrier supply is the quartz crystal
filter employed to select the 40,000-cycle carrier after frequency
doubling. The transmission characteristic of this extremely selective
filter is shown in Fig. 5.

294

BELL SYSTEM TECHNICAL JOURNAL

FREQUENCY IN PER CENT
99 100 101

102

103

dl5

' \

■^,

J^ OF 1 %
/T BAND

38.8 39.2 39.6 40.0 40.4 40.8 41.2

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 5 — ^Transmission characteristic of carrier supply crystal filter.

Filters

The transmission characteristics of the carrier channels are deter-
mined largely by the filters and associated equalizers. The filters
principally affecting transmission are the band filters. Identical units
are employed at the sending and receiving ends. The transmission
and phase shift characteristics of one of these units are shown in Fig. 6.
These band filters are equalized to produce the desired squared band
characteristic.

The characteristics in the frequency region near the carrier (i.e., at
40,000 cycles) are shown on a large scale. This region is of particular
interest because it is here that the degree of success in the application
of the vestigial sideband method, for the purpose of insuring the satis-
factory transmission of the low music frequencies, is determined. If
for a given frequency interval above the carrier the phase change is
arranged to be equal and opposite to that of the same frequency
interval below the carrier, then in the action of demodulation the
demodulated current produced by the action of one sideband adds

TRANSMISSION LINES

295

itself arithmetically to that produced by the other sideband. It will
be noted that this desirable phase characteristic has been achieved
closely in the characteristics shown. If, in addition, the attenuation

14 18 22 26 30 34 38 42 46

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 6 — Transmission and phase characteristics of band filter.

loss in the filter is adjusted so that the sum of the regular and vestigial
sideband amplitudes corresponding to the low music frequencies is
substantially constant and equal to the amplitude of the frequencies at
midband, the desired flat transmission characteristic is assured.

As was noted previously, this action can be carried out only if the
phase angle of the receiving carrier is properly related to that of the
sideband frequencies and, in turn, to the carrier applied to the modu-
lator at the transmitting terminal. The curves shown in Fig. 7

LlJ
Q
2-4

ai
o

APPROXIMATE PHASE
-ANGLE OF RECEIVING

PHASE ANGLE ADJUSTED
FOR SIDEBAND ADDITION

CARRIER IN DEGREES

3P-

—

■^

0
25

-—

«^ PHASE ANGLE ADJUSTED
FOR SIDEBAND OPPOSITION

to p

If)
1-2

iu-4

WITHOUT LOW
FREQ. SUPPRESSION

AS USED WIT

LOW FREQ

-SUPPRESSIOr

D 10 20 30 40 5
FREQUENCY IN CYCLES

100

500 1000 5000

FREQUENCY IN CYCLES PER SECOND

10,000 20,000

Fig. 7 — Over-all transmission characteristics as a function of the phase relation of

the receiving carrier.

296

BELL SYSTEM TECHNICAL JOURNAL

illustrate the influence that the phase adjustment of the carrier
frequency has on the transmission of the lower frequencies in a system
of this kind.

The upper curve shows the transmission frequency characteristic
of one of the carrier channels measured from terminal to terminal
between distortionless lines, when the phase angle of the receiving
carrier is adjusted for its optimum value. Under these conditions the
vestigial sideband and normal sideband supplement each other in their
effects to produce substantially flat transmission. (The insert in-
dicates the sustained transmission toward zero frequency when the
40-cycle highpass filter is omitted from the circuit.) It may be noted
also that with this proper phase adjustment the full band transmission
characteristic provided is substantially flat within a fraction of a
decibel from 40 c.p.s. to 15,000 c.p.s. The lower curves indicate
successively what happens if the phase angle of the receiving carrier
is adjusted different amounts from the optimum adjustment. It may
be noted that for a 90-degree departure the transmission of a 40-cycle
tone over the carrier channel would suffer more than 12 db in com-
parison with a 1000-cycle tone.

Repeaters
As noted previously, the line circuit between Philadelphia and
Washington included five intermediate repeater points. A schematic
drawing of the apparatus installed at each point is shown in Fig. 8.

BUILDING

OUT
EQUALIZER

BASIC
EQUALIZER

REGULATING
NETWORK

AMPLIFIER

REGULATING
CONTROL

Fig. 8 — Schematic diagram of repeater station apparatus.

The amplifiers at these points, as well as those used at the transmitting
and receiving terminal, consisted of a new form of amplifier employing
the principle of negative feed-back. The principal virtues of am-
plifiers of this type are their remarkable stability with battery and
tube variations and great freedom from nonlinearity or modulation
effects. Each amplifier is supplemented at its input by an equalizer
designed to have its attenuation approximately complementary in
loss to that of the line circuit in a single section. The amplifiers
actually employed for the purpose were taken from a trial of a cable

TRANSMISSION LINES

297

carrier system described in a recent A. I. E. E. paper by A. B. Clark
and B. W. Kendall.^

The losses in the cable circuits do not, of course, remain absolutely
constant with time, and slow variations due to change of temperature
are compensated for by occasional adjustments of the variable equal-
izer arrangements provided. These adjustments were required only
infrequently; approximately at weekly intervals because in an under-
ground cable the temperature experiences only slow, seasonal
variations.

As noted, new repeater stations were established at two points. The
housing arrangements for one of these points, Abingdon, is shown in
Fig. 9. The equipment at this repeater point also included relays
remotely controlled from the nearest attended repeater station to
permit the repeaters to be turned on and off at will and the power
supply, which consisted of storage batteries, to be switched from the
regular to the reserve battery or either battery put on charge if
required.

Fig. 9 — Interior and exterior of special intermediate ri-poatcr
Abingdon, Md.

:.Latiou at

298

BELL SYSTEM TECHNICAL JOURNAL

Over-All Performance
While the system was set up specifically to provide transmission for
the demonstration into Washington on April 27, 1933, it was operated
over a period of sev^eral weeks and complete tests and measurements
were carried out for the purpose of gathering information on cable
carrier systems. The complete layout of apparatus and lines provided
between Philadelphia and Washington is shown in Fig. 10.

V) -2

a. -6

RIGHT CHANNEL
/

LEFT CHANNEL
/

.._jl._

«k,

■r^r^r^

s.,

/As

M-^ — ^

CENTER CHANNEL

—

t^^

100

500

1000

5000 10,000 20,000

FREQUENCY IN CYCLES PER SECOND

Fig. 1 1 (right) — Freqiicnc}' characteristics over carrier channels used between Phila-
delphia and Washington, D. C.

The over-all frequency transmission characteristics of the three chan-
nels that were set up are shown in Fig. 11. These curves differ from
those shown in Fig. 7, and include the complete high frequency line
circuit with its 150 miles of cable, repeaters, equalizers, and other
equipment. It may be seen that between the desired frequency
limits the circuit is substantially flat in transmission performance to
within ± 1 db. Various noise measurements made on the over-all
circuit indicated that the circuits fully met the requirements that had
been set up, and that the line and apparatus noise was inaudible in the
auditorium at Washington even during the weakest music passages.
The circuit also was found to be free from nonlinear distortion to a
satisfactory degree. Harmonic components generated when single-
frequency tones were applied to the channels at high volumes were
found with one unimportant exception to be more than 40 db below
the fundamental.

As a means of obtaining a further increase in volume range, which
was not actually required for this demonstration, tests were made
with a so-called predistortion-restoring technique. In this the higher
frequency components of the music were transmitted over the carrier
channels at a volume much higher than normal in relation to the
volume of the lower frequencies. By this means any noise entering
the carrier channels at frequencies equivalent to the higher music
frequencies is greatly minimized in effect.

^|i^v4%f{l]|^j

ACAOEMV Of MUSIC

I r-'^

r;;,;;o;:pi

lATOLL (g.

--1 r~]

! 23.8 1

^iTif^ir^hzf^^s^'^M)^--^

' ' ■ I i
■yh2r^^EFsSi>]----|^---ji]i>|-'

I I

1 I

1 i

I I
I I

:.yh^-

CABRIER
I '

I 822 I

i !

! I '

j I I I

' I I I I ' '

I i I I

LJ lJ LJ

0 H

(T| EQUALIZE

PREDISTORTING

- SYMBOLS

(VJ) VOLUME INDICATOR

RESTORING NETWORKS -^A^ VARIABLE

r' OR POTEN

lAI .40- 15,000 CYCLE AMPLIFIER
—IN— CARRIER AMPLIFIER

rir;! carrier transmitting

IILLJ TERMINAL

fTTri CARRIER RECEIVING
lED TERMINAL

Fig. 10 — Schematic diagram of circuit layout for 15-kc cliannel used for symphonic program demonstration.

TRANSFORMER

D— MICROPHONE

14>-L0U0 SPEAKER

/"~\ MONITORING

I f] HEAD RECEIVER

TRANSMISSION LINES

299

This predistortion is accomplished by including in the circuit at
the input to the modulator a network having relatively high loss for
the lower frequencies and tapering to low loss for the higher frequencies.
Its maximum loss is compensated for by adding in the circuit an
equivalent amount of additional amplification. The characteristics
of such a network are illustrated in Fig. 12. To restore the normal
volume relationships between the different tones and overtones a

26
24
22

20
18
16

COMPOSITE

1 1

. .

—

ol^>

<?•

—

■ —

LU
Q

212

SlO

0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 12 — Attenuation characteristics of "predistorting and restoring" networks.

restoring network having complementary transmission frequency
characteristics is, of course, included at the output of the receiving
circuit. It was found with this predistortion-restoring technique that
a volume range increase of something like 10 db could be obtained over
the circuits described.

There is available also another method which might have been
employed for obtaining a further increase in volume range. This
method, the so-called volume compression-expansion system, very
likely will be necessary if in the future it is desired to obtain such high
quality circuits on long routes where the carrier frequency range is
being used also for regular telephone message transmission or for other
purposes, and where the problem of freedom from noise and crosstalk

TRANSMISSION LINES

299

26
24
22

20
18
16

UI4

LLl

Q
212

to
Sio

-J

6
4

COMPOSITE

1 1

—

}yi-

■ —

4 5 6 7 8 9 10 II 12

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 12— Attenuation characteristics of "predistorting and restoring" networks.

300 BELL SYSTEM TECHNICAL JOURNAL

no doubt will be more serious than experienced in the Philadelphia-
Washington demonstration. Such a volume compression-expansion
system requires additional apparatus at the sending and receiving
terminals of the line circuit. At the sending end this apparatus is
used to raise in volume the weak passages of the music or other program
for transmission over the line circuits in order that the proper ratio
between the desired program and unwanted noises may be retained.
At the receiving terminal coordinating apparatus reexpands the com-
pressed volume range to the volume range originally applied to the
transmitting terminal.

In the demonstration, to provide supplementary control features
required by Dr. Stokowski at Washington for communicating with
the orchestra at Philadelphia, additional wire circuits were established
between these points. Order wire circuits also were provided for
communication between the terminals and repeater points to make
possible the location troubles if any should arise. Rather elaborate
switching means were included at the terminals to permit switching
the carrier channels to different microphones and to different amplifier
equipment at the loud speaker end. To take care of the contingency
of a cable pair failure, spare pairs of wires were made available to be
switched in at short notice. Fortunately, none of the reserve facilities
actually were required for the demonstration.

References

1. "Long Distance Cable Circuits for Program Transmission," A. B. Clark and

C. W. Green. A. I. E. E. Trans., v. 49, 1930, p. 1514-23.

2. "Thermal Agitation of Electricity in Conductors," J. B. Johnson. Phys. Rev.,

V. 32, 1928, p. 97.

3. "Thermal Agitation of Electric Charge in Conductors," H. Nyquist. Phvs. Rev.,

V. 32, 1928, p. 110.

4. "Carrier in Cable," A. B. Clark and B. W. Kendall. Elec. Engg., Julv 1933, p.

477-81. ^s - J -

System Adaptation*

By E. H. BEDELL and IDEN KERNEY

A communication system for the pick-up and reproduction in auditory
perspective of symphonic music must be designed prop)erly with respect to
the acoustics of the pick-up auditorium and the concert hall involved. The
reverberation times and sound distribution in the two auditoriums, the
location of the microphones and loud speakers, and the response-frequency
calibration of the system and its equalization are considered. These and
other important factors entering into the problem are treated in this paper.

WHEN the effect of music or the intelUgibiHty of speech is spoiled
by bad acoustics in an auditorium, the audience is well aware
that acoustics do play a most important part in the appreciation of the
program. One may not be conscious of this fact when the acoustical
conditions are good, but a simple illustration will show that the effect
still is present. Thus, of the sound energy reaching a member of the
audience as much as 90 per cent may have been reflected one or more
times from the various surfaces of the room, and only 10 per cent
received directly from the source of the sound.

In listening to reproduced sound in an auditorium or concert hall,
the effect of the room acoustics is perhaps even more important, for in
this case the audience does not see any one on the stage and must rely
entirely upon the auditory effect to create the illusion of the presence
there of an individual or a group. Imperfections in the reproduced
sound that are caused by defects in the acoustics of the auditorium
may destroy the illusion and be ascribed improperly to the reproducing
system itself.

In some types of reproduced sound, radio broadcast for example,
where the reproduction normally takes place in a small room, the
attempt is made to create the illusion that the listener is present at the
source.^' 2 In the case considered here, however, where symphonic
music is reproduced in a large auditorium, the ideal is to create the
illusion that the orchestra is present in the auditorium with the
audience. Since the orchestra is playing in one large room and the
music is heard in another, the acoustical conditions prevailing in both
must be considered.

* Sixth and final paper in the Symposium on Wire Transmission of Symphonic
Music and Its Reproduction in Auditory Perspective. Presented at Winter Con-
vention of A. I. E. E., New York City, Jan. 23-26, 1934. Published in Electrical
Engineering, January, 1934.

301

302

BELL SYSTEM TECHNICAL JOURNAL

Pick-Up Conditions
The source room is the auditorium of the American Academy of
Music in Philadelphia. This room has a volume of approximately
700,000 cubic feet, and a seating capacity of 3000. Measured reverber-
ation time curves for this auditorium, and preferred values ^-^ for a
room of this volume, are given in Fig. 1. It may be seen that with a

yj 1.0

1 - MEASURED VALUES, EMPTY

2- MEASURED VALUES, FULL AUDIENCE

3 -ACCEPTED OPTIMUM FOR DIRECT LISTENING
4-ACCEPTED OPTIMUM FOR MONAURAL PICK-UP

""^

""^•.^

—

' ■ -^^

--^

-~-lZ!

^-_

500 1000

FREQUENCY IN CYCLES PER SECOND

Fig. 1 — Reverberation characteristics of Academy of Music, Philadelphia, Pa.

full audience this room might be considered somewhat dead, but would
be considered generally satisfactory for pick-up either with or without
an audience. A floor plan of the Academy auditorium and stage,
showing the location of the three microphones used, is given in Fig. 2.
The microphone positions were selected after judgment tests using
several locations and are much nearer the orchestra than they would
be for single channel pick-up.^ The use of the microphones near the
orchestra results in picking up a high ratio of direct to reverberant
sound and thus reduces the effect of reverberation in the source room
upon the reproduced music. A high ratio of direct sound is desirable
in the present case also because of the use of three channels. The per-
spective effect obtained with three channels depends to a considerable
extent upon the relative loudness at the three microphones, and since the
change in loudness with increasing distance from the source is marked
for the direct sound only, and not for the reverberant, there would be

SYSTEM ADAPTATION

303

a definite loss in perspective effect if the microphones were placed at
a greater distance from the orchestra. This effect is discussed more
fully in another paper of this symposium.

^^~~~~~~~-|

if)

ORCHESTRA

1 STAGE SET

DIRECTORS

STAND

^-^^"^

bTAGE

Fig. 2 — Floor plan of Academy of Music, showing location of microphones.

With the microphones located close to the orchestra their response-
frequency characteristics will be essentially those given by the normal
field calibration, since relatively little energy is received from the sides
and back. For a distant microphone position it would be necessary
to use the random incidence response characteristic, which differs from
the normal because of the variation in directional selectivity of the
microphones as the frequency varies. This difference in response
characteristic depends upon the size of the microphone and may
amount to as much as 10 db at 10,000 c.p.s. It may be pointed out
here that this difference in response is one factor frequently overlooked
in the placement of microphones.

In addition to the three microphones regularly used, a fourth was
provided to pick up the voice when a soloist accompanied the orchestra.
In this case only the two side channels were used for the orchestra, the
voice being transmitted and reproduced over the center channel. The
solo microphone was so shielded by a directional baffle that it responded
mainly to energy received from a rather small, solid angle. This
arrangement permitted independent volume and quality control for
the vocal and orchestral music.

The Concert Hall
The music was reproduced before the audience in Constitution Hall
in Washington, D. C. This hall has a volume of nearly 1,000,000

304

BELL SYSTEM TECHNICAL JOURNAL

cubic feet, and a seating capacity of about 4000. A floor plan of the
auditorium showing the location of the loud speakers and of the
control equipment is given in Fig. 3. The loud speakers are placed so
that each of the three sets radiates into a solid angle including as nearly

MEZZANINE

G''

DIRECTORS BOX
WITH CONTROL
EQUIPMENT

ORCHESTRA

Fig. 3 — Floor plan of Constitution Hall, Washington, D. C, showing locations of

loud speakers.

as possible all the seats of the auditorium. Figure 4 shows the rever-
beration-frequency characteristics of Constitution Hall. The values
given by the curve for the empty hall were measured through the use
of the three regular loud speakers and several microphone positions in the
room. The values for the hall with an audience present were calcu-
lated from known absorption data for an audience, and the optimum
values are taken from accepted data for an auditorium of the volume
of this one.^ The reverberation times were considered satisfactory and
no attempt was made to change them for this demonstration. The

SYSTEM ADAPTATION

305

reverberation time measurements for both Constitution Hall and the
Academy of Music were made with the high speed level recorder.^
This instrument measures and plots on a moving paper chart a curve

O 3.5

1 - MEASURED VALUES, EMPTY
2 -WITH FULL AUDIENCE
3 -ACCEPTED OPTIMUM

\ \
\ \ \

\ N

>s,^ X

\^*^

^^3

--,

2"^

l^rrrr^

-4

-—

100 200 500 1000 2000 5000

FREQUENCY IN CYCLES PER SECOND

Fig. 4 — Reverberation characteristics of Constitution Hall.

the ordinate of which is proportional to the logarithm of the electrical
input furnished to it. When used in connection with a microphone
for reverberation time measurements, curves are obtained showing the
intensity of sound at the microphone during the period of sound decay.
The rates of decay, and hence the reverberation times, are obtainable
immediately from the slopes of these recorded curves and the speed
of the paper chart.

Calibration of the System
In calibrating the system, a heterodyne oscillator connected to the
loud speakers through the amplifiers was used,. The oscillator was
equipped with a motor drive to change the frequency, and as the
frequency was varied through the range from 35 to 15,000 c.p.s. the
sound was picked up with a microphone connected to the level recorder.
Continuous curves of microphone response as a function of frequency
thus were obtained for several positions in the auditorium, and for each
channel independently. These response curves provided a check on a
uniform coverage of the audience by each loud speaker, and also
provided data for the design of the equalizing networks required to

306 BELL SYSTEM TECHNICAL JOURNAL

give an over-all flat response-frequency characteristic. If the system,
including the air path from the loud speakers to one position in the
auditorium, is made flat, it will not, in general, be flat for other posi-
tions or for other paths in the room. This variation in characteristic
is due partly to the variation in the ratio of direct to reverberant
sound, and partly to the fact that the sounds of higher frequency are
absorbed more rapidly by the air during transmission.^- ^ This latter
effect is of considerable importance; it depends upon the humidity and
temperature of the air, and may cause a loss of more than 10 db in
the high frequencies at the more distant positions in a large auditorium.
Some compromise in the amount of equalization employed therefore
is necessary. Probably the most straightforward procedure would be
to design the networks according to the response curves obtained with
the microphones near the loud speakers. This would insure that for
both the response measurements and the pick-up the microphone
characteristics would be the same, and any deviation from a uniform
response in the microphones would be corrected for in this way, along
with variations in the loud speaker output. This procedure was
modified somewhat for the case under discussion, however, because
by far the greater portion of the audience was at a distance from the
stage such that they received a relatively large ratio of reverberant
sound, and it was believed that a better effect would be achieved by
equalizing the system characteristic in accordance with response
measurements taken at some distance from the loud speakers.

Control Equipment

In addition to the equalizing circuits used to obtain a uniform re-
sponse characteristic, two sets of quality control networks which could be
switched in or out of the three channels simultaneously were employed.
One set modified the low frequencies as shown at A, B, and C of Fig. 5,
while the other gave high frequency characteristics as shown at D, E,
F, and G. These latter networks permitted the director to take
advantage of the fact that the electrical transmission and reproduction
of music permits the introduction of control of volume and quality
which can be superimposed on the orchestral variations. Quality of
sound can be divorced from loudness to a greater degree than is
possible in the actual playing of instruments, and the quality can be
varied while the loudness range is increased or decreased. Electrical
transmission therefore not only enlarges the audience of the orchestra,
but also enlarges the capacity of the orchestra for creating musical
effects.

The quality control networks and their associated switches were

SYSTEM ADAPTATION

307

mounted in a cabinet (Fig. 6) at the right side of the director's position.
Continuously variable volume controls for the three channels were
mounted on a common shaft and housed in the center cabinet of Fig. 6.

10
0

A
B

, ^

'-^

y^^

-10

20
-.10

'^•

20 50 100 500 1000 5000 10,000 20,000

FREQUENCY IN CYCLES PER SECOND

Fig. 5 — Transmission characteristics of quality control networks used in the Phila-
delphia-Washington experiment.

A separate control for the center channel was provided when that was
used for the soloist. In addition to the high quality channels certain
auxiliary circuits were supplied to aid the smoothness of performance.
Supplementing the order wire connecting all technical operators, a
monitor circuit was provided in the reverse direction. The microphone
was located on the cabinet before the director, and loud speakers were
connected in the control rooms and on the stage with the orchestra,
enabling the control operator to hear what went on in the auditorium
and allowing the director to speak to the orchestra. Two useful
signal circuits were employed; one giving the orchestra a "play" or
"listen" signal, and at the same time connecting either the auditorium
or the orchestra's loud speakers, respectively; the other being a

Fig. 6 — Cabinets housing quality control networks and providing communication

facilities for operation.

308 BELL SYSTEM TECHNICAL JOURNAL

"tempo" signal to the assistant director leading the orchestra that
could be operated during the rendition of the music. The switches
for the auxiliary circuits and the order wire subset are shown at the
control operator's position at the left in Fig. 6.

That a reproducing system may have quite different characteristics
in different auditoriums is well illustrated in the case of the two halls
considered here. From Fig. 3 it may be seen that in Constitution
Hall the stage is built into the auditorium itself, and that there is no
back stage space. The Academy of Music, however, has a large
volume back stage. When the orchestra plays in the Academy the
reflecting shell shown in Fig. 2 is used to concentrate the radiated
sound energy toward the audience. When the reproducing system
was set up in the Academy the shell could not be used because of the
stage and lighting effects desired, and a large part of the energy radi-
ated by the loud speakers at the low frequencies was lost back stage.
The loss of low frequency energy is attributable partly to the fact that
the loud speakers cannot well be made as directional for the very low
frequencies as for the higher. The loss amounts to about 10 db at
35 c.p.s., and becomes inappreciable at 300 c.p.s. or more, as measured
in comparable locations in the two auditoriums. This difference in
characteristics emphasizes the fact that for perfect reproduction the
acoustics of the auditorium must be considered as a part of the system,
and that in general the equalizing networks must have different charac-
teristics for different auditoriums.

References

1. "Acoustics of Broadcasting and Recording Studios," G. T. Stanton and F. C.

Schmid. Jour. Acous. Soc. Am., v. 4, No. 1, part 1, July 1932, p. 44.

2. "Acoustic Pick-Up for Philadelphia Orchestra Broadcasts," J. P. Maxfield. Jour.

Acous. Soc. Am., v. 4, No. 2, Oct. 1932, p. 122.

3. "Optimum Reverberation Time for Auditoriums," W. C. MacNair. Jour. Acous.

Soc. Am., V. 1, No. 2, part 1, Jan. 1930, p. 242.

4. "Acoustic Control of Recording for Talking Motion Pictures," J. P. Maxfield.

Jour. S. M. P. E., V. 14, No. 1, Jan. 1930, p. 85.

5. "A High Speed Level Recorder for Acoustic Measurements," E. C. Wente, E. H.

Bedell, K. D. Swartzel, Jr. Unpublished paper presented before the Acous.
Soc. Am., May 1, 1933.

6. "The Effect of Humidity Upon the Absorption of Sound in a Room, and a Deter-

mination of the Coefficients of Absorption of Sound in Air, V. O. Knudson.
Jour. Acous. Soc. Am., v. 3, No. 1, July 1931, p. 126.

7. "Absorption of Sound in Air, in Oxygen, and in Nitrogen — Effects of Humidity

and Temperature," V. O. Knudson. Jour. Acous. Soc. Am., v. 5, No. 2,
Oct. 1933, p. 112.

Abstracts of Technical Articles from Bell System Sources

Effects of Rectifiers on System Wave Shape} P. W. Blye and H. E.
Kent. Operation of mercury arc rectifiers generally results in in-
creased harmonic currents in the rectifier supply circuits and may re-
sult in increased harmonic voltages. While these harmonics usually
are not serious from the standpoint of the power system, they may
result in interference to communication circuits exposed to the power
circuits. This paper presents a method of computing these harmonic
voltages and currents, and discusses methods of coordinating telephone
systems and a-c. power systems supplying rectifiers.

Joint Use of Poles with 6,900-Volt Lines} W. R. Bullard and D.
H. Keyes. a plan has been developed for joint occupancy of poles by
power and telephone circuits in the Staten Island, N. Y. area, involv-
ing 6,900-volt distribution. The aim of this plan is to secure to the
public and to the power and telephone companies over-all safety,
convenience, and economy. Results of this cooperative study of
joint use are presented in this paper.

Sound Film Printing — //.'' J. Crabtree. The production of
sound-film prints from variable density negatives by the Model D
Bell & Howell printer has been studied from the point of view of high-
frequency response and uniformity of product. The account of this
study, begun in Part I, is continued here, with particular reference to
the degree of influence of slippage on the high-frequency response,
occasioned particularly by non-conformity of the perforation pitch
of the negative and positive films. It is found that to improve print-
ing conditions in practice, it is first necessary to achieve consistency in
the pitch of the processed negative and positive materials and to make
the pitch of the processed negative 0.0004 inch less than that of the
positive raw stock.

The Determination of the Direction of Arrival of Short Radio Waves }
H. T. Friss, C. B. Feldman, and W. M. Sharpless. In this paper
are described methods and technique of measuring the direction with

1 Elec. Engg., January, 1934.
^ Elec. Engg., December, 1933.
3 Jour. S. M. P. E., February, 1934.
*Proc. I. R. E., January, 1934.

309

310 BELL SYSTEM TECHNICAL JOURNAL

which short waves arrive at a receiving site. Data on transatlantic
stations are presented to illustrate the use of the methods. The meth-
ods described include those in which the phase difference between two
points constitutes the criterion of direction, and those in which the
difference in output of two antennas having contrasting directional
patterns determines the direction. The methods are discussed first
as applied to the measurement of a single plane w^ave. Application
to the general case in which several fading waves of different directions
occur then follows and the difficulties attending this case are discussed.

Measurements made with equipment responsive to either the hori-
zontal or the vertical component of electric field are found to agree.

The transmission of short pulses instead of a steady carrier wave is
discussed as a means of resolving the composite wave into components
separated in time. More detailed and significant information can
be obtained by this resolving method. The use of pulses indicates
that (1) the direction of arrival of the components does not change
rapidly, and (2) the components of greater delay arrive at the higher
angle above the horizontal. The components are confined mainly to
the plane of the great circle path containing the transmitting and
receiving stations.

A method is described in which the angular spread occupied by the
several component waves may be measured without the use of pulses.

Application of highly directional receiving antennas to the problem
of improving the quality of radiotelephone circuits is discussed.

Electron Diffraction and the Imperfection of Crystal Surfaces.^ L. H.
Germer. Bragg reflections are obtained by scattering fast electrons
(0.05A) from the etched surfaces of metallic single crystals. The
surfaces studied are a (100) face of an iron crystal, (111) face of a
nickel crystal and (110) face of a tungsten crystal. In each case the
reflections occur accurately at the calculated Bragg positions with no
displacement due to refraction. A given reflection is found, however,
even when the glancing angle of the primary beam differs considerably
from the calculated Bragg value — by over 1.0° in some cases — so that
several Bragg orders occur simultaneously. The accuracy with which
this glancing angle must be adjusted is a measure of the degree of
imperfection of the crystal. From the electron experiments, estimates
are made of the widths at half maximum of electron rocking curves.
These widths are 0.8° for the iron crystal, 1.5° for the nickel crystal
and somewhat over 1.0° for the tungsten crystal. X-ray rocking curves
for these same crystals are much narrower, although the observed

^ Phys. Rev., December 15, 1933.

ABSTRACTS OF TECHNICAL ARTICLES 311

widths vary considerably with the treatment of the surfaces. It is
concluded that the values obtained from the electron measurements
apply to projecting surface metal only, and that the degree of misalign-
ment is much greater at the surface than deep down within the crystal.
Furthermore, even the x-rays [Mo Ka radiation — 0.71 A] are not
sufficiently penetrating to yield values certainly characteristic of these
metal crystals.

Mutual Impedance of Grounded Wires Lying on the Surface of the
Earth when the Conductivity Varies Exponentially with Depth.^ Marion
C. Gray. This paper presents a formula for the mutual impedance of
any insulated wires of negligible diameter lying on the surface of the
earth and grounded at their end-points, on the assumption that the
conductivity of the earth varies exponentially with depth. Various
special cases are briefly discussed.

Signals and Speech in Electrical Communication. "^ John Mills.
This book is written by a member of the technical staff of Bell Tele-
phone Laboratories who is well-known for his text on "Radio Com-
munication" (1917) and the more popular presentations of "Within
the Atom" (1921) and "Letters of a Radio-Engineer to His Son"
(1922). In this book he presents for the general reader a synthesis of
the electrical arts of communication in terms of their general funda-
mental principles. In separate chapters, which are discrete essays in
popular and semi-technical language, the fundamental principles of
dial operation, transmitters and receivers, loading coils, repeaters,
multi-channel or carrier systems, and transoceanic radio-telephony are
graphically expounded. The entertaining treatment of engineering
achievements in allied fields of the sound picture, broadcasting, tele-
vision, stereophonic reproduction and the teletypewriter, will intrigue
the layman and assist him in acquiring a general understanding of
these highly technical developments.

Some Earth Potential Measurements Being Made in Connection with
the International Polar Year.^ G. C. Southworth. For several years
the Bell System has been studying the relation between radio trans-
mission and earth potential disturbances. A paper dealing with this
subject was published in 1931. Prompted by the needs of the Inter-
national Polar Year, together with the prospect that further work
would throw additional light on the nature of radio transmission, the
work was extended somewhat in 1932.

^Physics, January, 1934.

' Published by Harcourt Brace and Company, New York, N. V., 1934.

» Proc. I. R. E., December, 1933.

312 BELL SYSTEM TECHNICAL JOURNAL

It is expected that useful correlation will be found between the nor-
mal earth potential effects which occur day after day during undis-
turbed periods and the corresponding diurnal and seasonal variation of
radio transmission. It seems entirely probable, for instance, that
earth potentials are but the terrestrial manifestations of certain changes
taking place in the Kennelly-Heaviside layer which may not be found
by other methods.

This paper is intended to serve mainly as a progress report outlining
briefly the methods and scope of the work and showing the type of
data being obtained. It leaves to a later date most of their correlation
and their interpretation. The data here presented are in a conven-
tional form used by other investigators for many years. Their value
lies mainly in their extent and in the rather wide range of circumstances
under which they were obtained.

Investigation of Rail Impedances.^ Howard M. Trueblood and
George Wascheck. Measurements of impedance made on five sizes
of rails and on two types of bonds are reported in this paper; the inves-
tigation covered a range of current per rail of 20 to 900 amperes, and
frequencies of 15 to 60 cycles per second. Results are given in a form
convenient for engineering use, and include information for applying
corrections for bond impedance and for temperature.

^ Elec. Engg., December, 1933.

Contributors to this Issue

H. A. Affel, S.B. in Electrical Engineering, Massachusetts Insti-
tute of Technology, 1914; Research Assistant in Electrical Engineering,
1914-16. Engineering Department and the Department of Develop-
ment and Research, American Telephone and Telegraph Company,
1916-34. Bell Telephone Laboratories, 1934-. Mr. Affel has been
engaged chiefly in development work connected with carrier telephone
and telegraph systems.

E. H. Bedell, B.S., Drury College, 1924; University of Missouri,
1924-25. Bell Telephone Laboratories, Acoustical Research Depart-
ment, 1925-. Mr. Bedell's work has had to do mainly with studies of
sound absorption and transmission and allied subjects in the field of
architectural acoustics.

Arthur G. Chapman, E.E., University of Minnesota, 1911. Gen-
eral Electric Company, 1911-13. American Telephone and Telegraph
Company, Engineering Department, 1913-19, and Department of
Development and Research, 1919-34. Bell Telephone Laboratories,
1934-. Mr. Chapman is in charge of a group engaged in developing
methods for reducing crosstalk between communication circuits, both
open wire and cable, and evaluating effects of crosstalk on telephone
and other services.

R. W. Chesnut, A.B., Harvard University, 1917; War Depart-
ment of French Government, 1917; U. S. Army, 1917-19. Engineer-
ing Department, Western Electric Company, 1920-25; Bell Telephone
Laboratories, 1925-. Mr. Chesnut has been engaged in the develop-
ment of carrier telephone and long-wave radio systems.

Harvey Fletcher, B.Sc, Brlgham Young University, 1907; Ph.D.,
University of Chicago, 1911; Instructor of Physics, Brlgham Young
University, 1907-08, and University of Chicago, 1909-10; Professor,
Brlgham Young University, 1911-16. Engineering Department,
Western Electric Company, 1916-25; Bell Telephone Laboratories,
1925-. As Acoustical Research Director, Dr. Fletcher Is In charge of
investigations in the fields of speech and audition.

Frederick S. Goucher, A.B., Acadia University, 1909; A.B.,
Yale University, 1911; M.A., Yale, 1912; Ph.D., Columbia University,
1917. Western Electric Company, 1917-18. University College,
London, 1919. Research Laboratories, General Electric Company,
Limited, North Wembley, England, 1919-26. Bell Telephone Labora-
tories, 1926-. Dr. Goucher has been engaged in a study of the carbon

microphone.

313

314 BELL SYSTEM TECHNICAL JOURNAL

Iden Kerney, B.S. in Communication Engineering, Harvard
Engineering School, 1923. Development and Research Department,
American Telephone and Telegraph Company, 1923-34. Bell Tele-
phone Laboratories, 1934-.

R. H. jVIills, S.B. in Electrical Engineering, Massachusetts Insti-
tute of Technology, 1916. Western Union Telegraph Company, 1916-
18. Western Electric Company, Transmission Development Branch,
1918-25. Bell Telephone Laboratories, Apparatus Development
Department, 1925-. Mr. Mills is responsible for the development of
carrier frequency filters for use in commercial communication systems.

E. O. ScRiVEN, B.S., Beloit College, 1906; Instructor, Fort Worth
University, 1906-08; S.M., Massachusetts Institute of Technology,
1911. Engineering Department, Western Electric Company, 1911-
25. Bell Telephone Laboratories, 1925-. Mr. Scriven is in charge of
the electrical design of special products apparatus other than radio.

W. B. Snow, A.B., Stanford University, 1923; E.E., 1925. Engi-
neering Department, Western Electric Compan}^ 1923-24. Acoustical
research, Bell Telephone Laboratories, 1925-. Mr. Snow has been
engaged in articulation testing studies and investigations of speech and
music quality.

J. C. Steinberg, B.Sc, M.Sc, Coe College, 1916, 1917. U. S.
Air Service, 1917-19. Ph.D., Iowa University, 1922. Engineering
Department, Western Electric Company, 1922-25; Bell Telephone
Laboratories, 1925-. Dr. Steinberg's work since coming with the
Bell System has related largely to speech and hearing.

A. L. Thuras, B.S., University of Minnesota, 1912; E.E., 1913.
Laboratory assistant with U. S. Bureau of Standards, 1913-16. Grad-
uate student in physics. Harvard, 1916-17. Bell Telephone Labora-
tories, 1920-. At the Laboratories, Mr. Thuras has worked on the
study and development of electro-acoustic devices and instruments.

E. C. Wente, A.B., University of Michigan, 1911; S.B. in Electrical
Engineering, Massachusetts Institute of Technology, 1914; Ph.D.,
Yale University, 1918. Engineering Department, Western Electric
Company, 1914-16 and 1918-24; Bell Telephone Laboratories, 1924-.
As Acoustical Research Engineer, Dr. Wente has worked principally
on general acoustic problems and on the development of special types
of acoustic devices.

VOLUME xm JULY, 1934

NUMBER 3

THE BELL SYSTEM

TECHNICAL JOURNAL

DEVOTED TO THE SCIENTinC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION

The Compandor — An Aid Against Static in Radio

Telephony— i?. C. Mathes and S. B. Wright . .315

The Effect of Background Noise in Shared Channel
Broadcasting — C. B. Aiken 333

Wide-Band Open- Wire Program System —

H. S. Hamilton 351

Line Filter for Program System — A. W. Clement , , 382

Contemporary Advances in Physics, XXVIII — The

Nucleus, Third Part— ^arZ K. Darrow . . . .391

Electrical Wave Filters Employing Quartz Crystals as

Elements— VT. P. Mason 405

Some Improvements in Quartz Crystal Circuit Elements

—F. R. Lack, G. W. Willard and I. E. Fair 453

A Theory of Scanning and Its Relation to the Charac-
teristics of the Transmitted Signal in Telepho-
tography and Television —

Pierre Mertz and Frank Gray 464

Abstracts of Technical Papers 516

Contributors to this Issue 520

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

NEW YORK

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The Bell System Technical Journal

July, 1934

The Compandor — An Aid Against Static in Radio Telephony *

By R. C. MATHES and S. B. WRIGHT

One of the important conditions which must be met by any speech
transmission system is that it should transmit properly a sufficient range
of speech intensities. In long-wave radio telephony, even after the speech
waves are raised to the maximum intensity before transmission, there
remain energy variations such that weak syllables and important parts of
strong syllables may be submerged under heavy static. The compandor
is an automatic device which compresses the range of useful signal energy
variations at the transmitting end and expands the range to normal at the
receiving end, thus improving the speech-to-noise ratio.

This paper deals with some of the fundamental characteristics of speech
waves and explains how the task of changing them for transmission over
the circuit and restoring them at the receiving end is accomplished. It
is also shown that raising the strength of the weaker parts of speech gives
these results: 1, the successful transmission of messages for a large per-
centage of the time previously uncommercial; 2, a reduction of the noise
impairment of transmission for moderate and heavy static during time
classed commercial; and 3, the ability to deliver higher received volumes
due to the improved operation of the voice controlled switching circuits.
In addition to these advantages, the compandor makes it possible to
economize on radio transmitter power in times of light static.

Introduction

WHEN the original New York-London long-wave radiotelephone
circuit was designed, it was recognized that radio noise would
often limit transmission, especially for the weaker voice waves. Ac-
cordingly provision was made for manually adjusting the magnitude of
the speech waves entering the radio transmitters to such a value as to
load these transmitters to capacity.^ While this treatment was very
effective in improving the average speech-to-noise ratio and in prevent-
ing the strong peaks of speech from overloading the transmitter, it was,
of course, unsuitable for following the rapidly varying amplitudes of
the various speech sounds.

The total range of significant intensities applied to the circuit is
in the order of 70 db, an energy ratio of 10 million to one. The manual
adjustments referred to above were succesful in reducing this range to
about 30 db. To further reduce this residual range an interesting

* Presented at Summer Convention of A. I. E. E., June, 1934. Published in
Electrical Engineering, June, 1934.

315

316 BELL SYSTEM TECHNICAL JOURNAL

device called the compandor has been developed. This device which
works automatically makes a further reduction of one-half in the resid-
ual db range so that the range transmitted over the circuit is then
only 15 db, an energy ratio of about 32 to one.

Speech Energy

Quantitative designation of speech intensity and hence of a range of
intensities is rendered difficult by the rapidly varying amplitude char-
acteristics of the various speech sounds. Devices called volume indi-
cators are used fairly extensively to indicate the so-called "electrical
volume" * of speech waves. A volume indicator is essentially a
rectifier combined with a damped d-c. indicating meter on which are
read in a specified manner the standard ballistic throws due to partly
averaged syllables at a particular speech intensity. These devices are
so designed and adjusted that they are insensitive to extremely high
peak voltages of short duration, but their maximum deflection is ap-
proximately proportional to the mean power in the syllable. It has
been found that, if commercial telephone instruments are used, the ear
does not detect amplifier overloading of the extremely high peaks of
short duration. Consequently, the volume indicator is a useful device
for indicating the noticeable repeater overloading effect of a voice wave.
These devices do not tell us much about the effect of the weaker volt-
ages in overriding interference or operating voice-operated devices
but they give a fairly satisfactory indication of loudness and possibil-
ities of interference into other circuits.

The sound energy that the telephone transmits consists of compli-
cated waves made up of tones of different pitch and amplitude. The
local lines and trunks connecting the telephone to the subscribers' toll
switchboard have little effect in changing the fundamental characteris-
tics of these waves but, on account of various amounts of dissipation,
the waves received at the toll switchboard are always weaker than those
transmitted by the telephone. Furthermore, the strength of signals
varies with the method of using the telephone, loudness of talking,
battery supply, and transmitter efficiency. The subscriber may be
talking over a long distance circuit from a distant city, in which case
the loss of the toll line further attenuates the received waves. Figure
1 t shows that the range of outgoing speech volumes as measured by a
volume indicator at the transatlantic switchboard at New York is
nearly 40 db for terminal calls. When via calls and variation in volume

* The term volume will be used through the rest of this paper to designate this
quantity and not as synonymous with loudness.-

t This curve is plotted on so-called probability paper, in which the scale is such
that data distributed in accordance with the normal law will produce a straight line.

THE COMPANDOR

317

of the individual talker are taken into account, it is even greater than
40 db.

Volume Range of a Telephone Circuit

There are two limits on the range of volumes which a system can
transmit. The upper limit of volume is set by the point at which

-25 -20 -15 -10 -5 0

DECIBELS RELATIVE TO MAXIMUM VOLUME

Fig. 1 — Volumes of 950 local subscribers at New York transatlantic switchboard,

January-April, 193 L

overloading appreciably impairs the signal quality or endangers the
life of the equipment. It is an economic limit set by the cost of build-
ing equipment of greater load capacity. The lower limit of volume is
set by the combination of the amount of attenuation and the amount
of interference in the system such that the signal should not be appre-

318 BELL SYSTEM TECHNICAL JOURNAL

ciably masked by noise. This also is ordinarily an economic problem
depending on the cost of lowering the attenuation or of guarding
against external interference. In some cases, however, this limitation
is a physical one. A striking case is that of radio transmission in which
we have no means of controlling the attenuation of the electromagnetic
waves in transit to the receiving station. They may arrive at levels
below those of thermaP'^ noise in the antenna and other receiving
apparatus. Thus, even in the absence of static there is a definite use-
ful lower limit to the received and hence the transmitted volume. In
such cases the problems raised by the spread in signal intensities become
a matter of particular importance. Radio telephony was therefore
one of the fields of use particularly in view for the development of the
device to be described.

Effect of Volume Control

Until recently the only method in use for reducing the range of
signal intensities on radio circuits was a special operating method for
constant volume transmission. At each terminal the technical oper-
ator, with the aid of a volume indicator, adjusted the speech volume
going to the radio transmitter to that maximum value consistent with
the transmitter load capacity.

Referring to Fig. 2, we have a diagram showing the normal relation
of input to output intensities of a zero loss transducer as given by the
diagonal line. Points ^max. and ^min. on this line indicate the ex-
treme values of signal intensities for sustained loud vowels covering a
volume range of 40 db. The effect of the volume adjustments made by
the technical operator is to bring all the applied volumes to a single
value indicated by point B in Fig. 2. The value of B could be any
convenient intensity. Here it is set at a value determined by trans-
mission conditions in the line between the technical operator's position
and the radio transmitter.

As the technical operator has reduced the strongest volumes 5 db
and increased the weakest volumes 35 db, the result of this volume
control is to increase the volume range which the circuit can handle by
40 db. It is possible to make this adjustment for two-way transmis-
sion in the case of radio circuits without danger of singing because of
the use of voice-controlled switching arrangements ^ which permit
transmission in only one direction at a time. By this method of opera-
tion volumes initially strong or weak are delivered to the distant re-
ceiving point with equal margins relative to interference and the trans-
mission capacity of the whole system is thereby improved.

THE COMPANDOR

319

^MAX

-20

Z-30

-60-

-50 -40 -30 -20

INPUT INTENSITY IN DECIBELS

Fig. 2 — Range contro!.

Intensity Range at Constant Volume

However, even with speech adjusted to constant volume at the
transmitting point there are large variations in signal intensity from
syllable to syllable and within each syllable. For example, the energy
of some consonants as compared with the stronger vowels is down about
30 db. The importance of the weaker sounds is brought out by the
fact that in the case of commercial telephone sets a steady noise 30 db
below the energy in the strongest parts of the speech syllables pro-
duces an appreciable impairment in transmission efficiency. It is
accordingly desirable to maintain transmission conditions such that
generally more than this range is kept free from the masking effect of
noise. This range of intensities within the syllable is also of importance
in the operation of the voice-controlled switches used in the radio
system. The sensitivity spread between a voice operated relay which

320 BELL SYSTEM TECHNICAL JOURNAL

just operates on the crests of loud syllables and one which operates
sufficiently well not to clip speech is also about 30 db.

Considering on Fig. 2 that the coordinates are in terms of the aver-
age r.m.s. value over a period of time small compared with the time of
a syllable, there is a spread of at least 30 db in signal intensity extend-
ing down from the maximum for each talker. Thus for the weakest
talker this spread is indicated by the bracket Y and for the strongest,
by X. Any other talker, as Z, falls somewhere in between. After
manual control of volume this spread of intensities is represented by
the bracket X', Y', Z' for all talkers. This residual spread makes
desirable a means for further compressing the range of intensities in
the speech signals so that the weaker parts of sound are transmitted
at a higher level without at the same time raising the peak values of
speech and so overloading the transmitter.

Types of Compression Systems

This problem can be approached in several ways. One, for in-
stance, is from the frequency distortion standpoint. As many of the
weaker consonants have their chief energy contribution in the upper
part of the speech band, a simple equalizer which relatively increased
the energy of the higher frequency consonants before transmission
and another which restored the frequency energy relations after trans-
mission should be found of value. Tests have confirmed this expecta-
tion to some degree. Unfortunately, the best type of equalizer de-
pends upon the type of subscriber station transmitter, so that in general
only a compromise improvement can be obtained.

Another general method of approach is that of amplitude distortion
in which the weaker portions of the syllable are automatically increased
in intensity in some inverse proportion to their original strength. The
manual control of volume described above may be considered the
genesis of this method. Early suggestions * included the use of an
auxiliary channel such as a telegraph channel for duplicating the con-
trol operations in the reverse sense at the receiving end, thus restoring
the original energy distribution. Another early suggestion along this
line was made by George Crisson of the American Telephone and
Telegraph Company. '^ If a voltage be applied to a circuit consisting
of a two-element vacuum tube (with a parabolic characteristic) in
series with a large resistance, the instantaneous voltages across the
tube are approximately the square root of corresponding voltages ap-
plied. ■ Thus a voltage originally 1/100 of the peak voltage can be
transmitted at an intensity of 1/10 of the peak or ten times its original
intensity. If the instantaneous energy is expressed on the logarithmic

THE COMPANDOR

321

or db scale, the energy range is then cut in half. Such a device may be
called an instantaneous compressor. At the distant end a circuit
which is simply the inverse of that at the transmitting end is used.
The output voltage is taken off of a low resistance in series with a
parabolic element, thus restoring the signal substantially to its original
form. This circuit may be called an instantaneous expandor. This
scheme was successfully tested in the laboratory but unfortunately
possesses a very serious limitation for practical application in the
telephone plant. This is due to the fact that, to properly maintain
the characteristics of the compressed signals, a transmission band
width without appreciable amplitude or phase distortion of about
twice the normal proved necessary.

The Compandor

The principle of the present device is the use of a rate of amplitude
control for the compressing and expanding devices intermediate be-
tween manual and instantaneous control which may be considered ap-
proximately as a control varying as a function of the signal envelope.^- ^
Such a modulation of the original signal in terms of itself does not
appreciably widen the frequency band width of the modified signal as
compared with the original signal. The transmitting device is called
the compressor; the receiving device, the expandor; and the complete
system, the compandor.

The functional behavior of a typical compressor may be considered
with reference to the simplified schematic circuit No. 1 of Fig. 3.

LINEAR I ] 3> t

ECTIFIER zf= =ir > Eo

T Tf t

AMPLIFIER

Fig. 3 — Compressor circuit No. 1.

322 BELL SYSTEM TECHNICAL JOURNAL

This circuit is of the forward-acting type; that is, the control energy is
taken from the line ahead of the point of variable loss. The variable
loss consists of a high impedance pad connected in the circuit through
two high ratio transformers Ti and Ti. The high resistances Ri and R2
are shunted by a pair of control tubes connected in push-pull. The
push-pull arrangement is desirable for two reasons. It reduces the
even order non-linear distortion effects caused by the shunt path on
the transmitted speech and it balances out the control impulse and
un filtered rectified speech energy from the control path which might
otherwise add distortion to the speech. The impedances of these
tubes are controlled by the control voltage Eg, which is roughly pro-
portional to the envelope of speech energy and which is derived from
the line through a non-linear or "rooter" * circuit, a linear rectifier
and a low-pass filter which may have a cutoff frequency in the range 20
cycles to 100 cycles. In the following analysis it is assumed that the
delay due to this filtering is negligible:

Let El = r.m.s. speech voltage at input
and £2 = r.m.s. speech voltage at output in same impedance

Re = a-c. impedance of control tubes.

Now if Re is kept small compared to the pad impedance, we have
approximately

E2 = kiEiRc. (1)

Let Eg he the control voltage applied to the grids of the control
tubes. With the plate voltage Eb just neutralized by the steady bias-
ing grid voltage Ec, then only Eg may be considered as determining
the space current and we may assume ideally that the space current

Ib = kiEG'.
Then

P cLEb dEc 1 .^x

^'--dTB-^-dTB-hE^^' ^2)

wheres 5 is determined by tube design and the ^s are constants for
constant ^l tubes. For variable /x tubes equation (2) can be used to
set requirements on the tube design.
FrorrL (1) and (2)

^^ = w^- '^)

Now let the rooter be a non-linear circuit such that the instantan-
eous voltage is the tth root of £1. After rectification and filtering we
* So called because the output is a root of the input; see equation (4).

THE COMPANDOR

323

shall have approximately
From (4) and (3) we have

\{ t = s = n

Eg = hEi'".

E2 = KEi"\

(4)

(5)

(6)

Now if the input voltage be increased by a factor x, the input
increment in db will be 20 log x. The new output will be

£2' = KixEiY'".

The increment in output in db will be

20 log^ = 20 log x'l"

-C-2

log x.

The ratio of the output increment to the input increment in db is l/w
and the device is said to have a compression ratio of l/n. In other
words, the per cent change in relative speech voltages in passing through
the compressor is the same at all points in the intensity range. In the
general form of this circuit, t and 5 need not be equal to secure a particu-
lar value of l/n.

In Fig. 4, Compressor Circuit No. 2 is shown, a backward-acting
type of circuit. In this circuit the control tubes can be used to per-

■AAA/

■VvV

AAA/

Fig. 4 — Compressor circuit No. 2.

324

BELL SYSTEM TECHNICAL JOURNAL

form the function of the rooter in circuit No. 1 when s = t = n. We
may write for this circuit

Eq = kiE^Rs,
Eo = kiE^,

1 1

Rb =

E, =

kzE^-^ hE,^-^ '

kiE,

(7)

which is the same as equation (6) for circuit No. 1.

In Fig. 5 is shown the Expandor Circuit. If the resistances r are

LINEAR
RECTIFIER

Fig. 5 — Expandor circuit,
kept small compared with those of the control tubes, we may write

R. '

= kiEj,

jti

hE,^-'

k^-,^-'

= KE^Ey-

-1 = KE^

(8)

This relation is just the inverse of that given in equations (6) and
(7). The increment ratio in db of output to input is n and the expan-
sion ratio may be said to be n. When a compressor and expandor
having the same value of n in their indices are put in tandem, the final
output and input intensity ranges are the same. However, between
the compressor and expandor the range of signal intensities, whose

THE COMPANDOR 325

rate of change is not faster than the usual syllabic envelope, is 1/w in
terms of db. In terms of voltage ratios the intermediate signal in-
tensities are proportional to the square root of their original values if n
equals 2, the cube root if n equals 3, etc.

The ideal relations postulated above cannot all be met in the
physical design of the circuits. The indices s and t must be the dy-
namic characteristics of the tube and circuit and can be held to constant
value only over limited ranges of operation. Equation 2 is only ap-
proximately true as some space current is permitted to flow when no
speech is passing; otherwise, impractical values of control impedances
would be involved. However, they do serve to illustrate the func-
tional operation and can be approximated sufficiently well in com-
mercial equipment for useful amounts of compression and expansion.
Figure 6 shows experimental steady-state input versus output charac-
teristics for devices built to have a compression ratio of 1/2 and an
expansion ratio of 2.

The compressor is seen to operate substantially linearly over a 45
db range of inputs and the expandor over a 22.5 db range. This is
about as much range as can be secured conveniently from a single
stage of vacuum tubes. As such ranges would be entirely insufficient
to handle the seventy odd db range at speech intensities, it is necessary
to control volumes to a given point before sending through these
devices, rather than compress or expand first and then control. The
range is adequate, however, to take care of the range of signal intensi-
ties for commercial speech at constant volume.

Effect of Compandor

The compressor curve of Fig. 6 indicates that, when the input is
15 db above 1 milliwatt, the compressor gives no gain or loss. If
the levels are adjusted so that this point corresponds to the intensity
at point B on Fig. 2, then the line 5C indicates the controlled intensities
corresponding to the assumed 30 db spread of speech controlled to
constant volume. The new range of intensities as indicated by the
bracket X" Y" Z" is now finally reduced to about 15 db. Tests show
that a volume indicator on the output of the compressor reads from 1 to
2 db higher than on uncompressed speech at its input. Compressed
speech sounds slightly unnatural but the effects of compression upon
articulation in the absence of noise are negligible.

In considering the action of the expandor it is important to note
that all of the improvement in signal-to-noise ratio is put in by the
compressor. Considering any narrow interval of speech the insertion
of the expandor does not change the signal-to-noise ratio. The de-

326

BELL SYSTEM TECHNICAL JOURNAL

sirability of using it depends on other reasons. First, it restores the
naturalness of the speech sounds. Second, the apparent magnitude
of the noise is greatly reduced since noise comes in at full strength only
when speech is loudest and is reduced by the loss introduced by the
expandor at times when the energy is low between syllables. When
no speech is being transmitted, noises up to a certain limit, which

5 -10

-20

-25

COMPRESSOR.

/expandor

-30

-20 -15 -10 -5 0 5

INPUT IN DECIBELS REFERRED TO 1 MILLIWATT

Fig. 6 — Experimental input vs. output characteristics (1000 cycles steady state).

corresponds to the maximum energy in received speech, are reduced
in varying amounts from about 20 db to zero depending on their value.
When speech is present the effect of the expandor is determined by
the sum of the instantaneous speech and noise voltages, so that the
effect on the noise, whether it is large or small, is determined largely
by the existing speech intensity. For a circuit having somewhere
near the limit of static, the use of the compandor allows on the average
5 db more noise than when it is not used. When the noise is less than

THE COMPANDOR 327

this limit, somewhat greater improvements are obtained from the
compandor, ranging up to at least 10 db.

The particular values of compression and expansion ratio were
chosen initially for the relative ease in the design of the system with
commercially available vacuum tubes whose characteristics closely
approximated a parabola. Tests of the equipment have shown that
this degree is sufficient for present telephone circuit intensity range
requirements. Increasing the amount of compression is limited by
increase in quality distortion and by increased variation in the in-
tensity of radio noise as heard by the listener. A noise which is con-
stant at the input to the expandor varies on the output as the speech
intensity changes. Also variations in attenuation equivalent between
the compandor terminals are multiplied by the expandor. Herein
lies a reason for having a constant compression and expansion ratio
over the working range. If it were different at different intensities,
attenuation changes would distort the reproduced speech as well as
appearing as a somewhat increased change in intensity. This change
in intensity is n times the attenuation change in front of the expandor
in db.

The degree of compression may obviously be controlled in a variety
of ways: such as, using different values for the indices 5 and /, applying
control voltages upon more than one variable stage in tandem, the use
of variable jj. vacuum tubes, etc. The circuits as shown use variable
shunt control for the compressor and variable series control for the
expandor. Either or both may be changed to the other by inverting
the polarity of the control potential and properly designing the rectifier
characteristics of the control circuits.

There are two major sources of possible speech distortion which
must be considered in the design and use of these devices in addition
to those ordinarily present. The first is due to the non-linear char-
acteristics of the vacuum tubes used for controlling. The even order
distortion terms are largely balanced out by using two tubes in a
push-pull arrangement. The remaining distortion is minimized
by having speech pass through the control tubes at a sufficiently low
level. In the operating ranges for the device shown on Fig. 6, the
harmonics of a single-frequency tone are 30 db or more below the
fundamental.

The second major source of distortion is the time lag in the control
circuits due to the presence of the filters after the linear rectifier.
However, with a complete compandor circuit using the compressor
circuit No. 1, it was found on careful laboratory tests with expert
listeners that it was almost impossible to distinguish whether the device

328

BELL SYSTEM TECHNICAL JOURNAL

was in or out of circuit. Furthermore, distortion of this type is largely
eliminated when compressor circuit No. 2 is used. In that case it will
be noted that, if the two terminals are connected by a substantially
distortionless transmission system, the identical control circuits of
the two devices receive identical operating voltages. As the gain
changes put in are reciprocal and occur now with equal time lag, the
deviations from ideal compression are virtually counterbalanced by the
inverse deviations from ideal expandor action. In Fig. 7 are shown

A-1
A-2
A-3

B-1
B-2
B-3

C-1
C-2
C-3

\. INPUT TO COMPRESSOR 2. OUTPUT OF COMPRESSOR 3. OUTPUT OF EXPANDOR
i

■^^^^ m/IjVV\'V\' ^-nv^^ai ^^^\^f^ WM^A

'NAAAyv/'WV v^ A'v-WVw '\r*AfW\/V^

■/\/^/^v/vVy^^M/"'v^¥A/\^^

vVwV-'-'W vA W^'A/WVV^^'

0.02 0.03

TIME IN SECONDS

0.04

0.05

Fig. 7 — Operation of compandor on beginning of word "bark." A. Compressor
circuit No. 1. B. Compressor circuit No. 2 with low-pass filter in control circuit.
C. Compressor circuit No. 2 without filter.

oscillograms taken of the first part of the word "bark." Each record
shows the intensity changes before the compressor, between the com-
pressor and expandor and on the output of the expandor.

Application to Transatlantic Circuit

A compandor system has been in service on the New York-London
long-wave radiotelephone circuit since about July 1, 1932. At first
compressor circuit No. 1 was used, and later a change was made to
compressor circuit No. 2. Figure 8 is a photograph of the experimental
installation at New York. It occupies about five feet of standard
relay rack space. The blank panel shown in the photograph indicates
the saving of apparatus resulting from the change to compressor
circuit No. 2. Figure 9 is a schematic diagram showing the method of
inserting the compressor and expandor in the radio telephone terminals
at each end of the circuit. Since the two ends are similar, only one

THE COMPANDOR

329

end is shown. The compandor circuits are indicated in their relation
to the subscriber, the toll switchboard, the vodas and privacy ap-
paratus, and the radio transmitter and receiver. ^ A meter located at
the point designated A would indicate the full range of applied volumes,
at B, the controlled volumes and at C, the compressed speech signals.

'^^'

t -u '

! 1>

* 1

1 '

^ ':

1" ii# 1

^'^^^^^^^^^■iM|^HMiH9HH|^BiHM^H <^J

15.

Fig. 8 — Experimental installation of compandor at New York.

When the United States subscriber talks, electrical waves set up
by his voice pass over a wire line to the toll switchboard. They then
divide in a hybrid set ; part of the energy is dissipated in the output of
a receiving repeater and part is amplified by a transmitting repeater
whose gain is controlled by noting the reading of a volume indicator at

330

BELL SYSTEM TECHNICAL JOURNAL

B and adjusting a potentiometer ahead of the transmitting repeater.
The waves then act on the vodas which consists of ampUfier-detector,
delay circuit and relays for switching the transmission paths in such a
manner as to prevent echoes, singing and other effects. When in the
transmitting condition, the vodas is arranged to have zero loss so that
the waves impressed on the compressor are practically the same as at
B. The waves put out from the compressor are then sent through the
privacy apparatus, the output of which is then sent over a wire line
to the radio transmitter. The radiated waves are picked up by the
distant radio receiver, amplified and transformed into voice-frequency
energy which passes over a wire line to the terminal at the distant end.
The path of received waves in either terminal may be traced in the
lower branch of the circuit shown on Fig. 9. After being made intel-

TRANSMITTING
REPEATER

UNITED

STATES

SUBSCRIBER

1^-

— a5cr| m^ —

-^mJ \s£u —

TOLL
SWITCHBOARD

HYBRID
SET

NETWORK

EXPANDOR

]£-:

COMPRESSOR

RECEIVING
REPEATER

LINE

RADIO ^
TRANSMITTER

WIRE
LINE

AUTOMATIC
VOLUME
CONTROL

RADIO ^^
RECEIVER

Fig. 9 — Compandor applied to one end of a radio telephone circuit.

ligible by passing through the receiving privacy device, the compressed
incoming waves are sent through the vodas into an automatic volume
control and then into the expandor. The expanded waves are sent
through a receiving repeater from whose output the amplified waves
pass into the hybrid set, part being dissipated in the network and the
other part going through the toll switchboard to the subscriber. Due
to imperfect balance between the subscriber's line and the network, a
portion of the received energy is transmitted across the hybrid set and
amplified by the transmitting repeater. This echo might operate the
transmitting vodas under certain conditions. For this reason a po-
tentiometer is inserted in the receiving branch of the circuit so as to
reduce the echo, and consequently the received volume, so that
false operation of the transmitting vodas is prevented.

THE COMPANDOR 331

Results of Compandor Operation

The effectiveness of the compandor in service depends not only on
its abiUty to reduce noise but also on its relation to the other character-
istics of the circuit. Tests in the laboratory and on the long-wave
transatlantic circuit have indicated that the presence of the compandor
does not affect the quality appreciably, provided compressor circuit
No. 2 is employed and provided the compression in the circuit itself
is not serious. Delay distortion can be tolerated up to about the same
amount as when no compandor is used. Frequency changing for
privacy purposes is not materially affected by the compandor.

The expander increases the transmission variations in the circuit
exactly as it increases the voltage range of the waves applied to it.
It is therefore necessary to guard against excessive variations in the
overall circuit including the wire line extensions as well as the radio
links. At the New York terminal there has been installed an automatic
volume control operated from received speech signals which performs
this function.

The received volume is limited by incoming waves which do not
operate the receiving side of the vodas but which return as echoes from
the land line to cause false operation of the transmitting side. The
compressor increases these weak waves so that they are better able to
operate the receiving side of the vodas, and the expandor effectively
increases the stronger waves relative to the weak. This results in more
received volume being delivered to the two-wire terminal than when the
compandor is not used. The overall improvement in volume delivered
to the subscriber varies with the noise, being greatest when the noise is
low.

Summary

The allowable increase of about 5 db in noise before reaching the
commercial limit increases the time when the circuit can be used for
service. The increased circuit time is greatest in the seasons of the
year when it is needed the most.

For conditions of moderate disturbances now classed as commercial,
a reduction of the noise transmission impairment to very low values is
accomplished by the compandor.

The improvement in the vodas operation results in delivering sub-
stantially higher volumes to the subscribers.

The beneficial effect of the compandor might alternately be ap-
plied to a reduction of transmitter power.

332 BELL SYSTEM TECHNICAL JOURNAL

References

1. "The New York-London Telephone Circuit," S. B. Wright and H. C. Silent,

Bell System Technical Journal, Vol. VI, pp. 736-749, October, 1927.

2. "Speech Power and Its Measurement," L. J. Sivian, Bell System Technical

Journal, Vol. VIII, pp. 646-661, October, 1929.

3. "Thermal Agitation of Electricity in Conductors," J. B. Johnson, Physical Review,

Vol. 32, pp. 97-109, July, 1928.

4. "Thermal Agitation of Electric Charge in Conductors," H. Nyquist; presented

before the American Physical Society, February, 1927, and published in Physi-
cal Review, Vol. 32, pp. 110-113, July, 1928.

5. "Two-Way Radio Telephone Circuits," S. B. Wright and D. Mitchell, Bell

System Technical Journal, Vol. XI, pp. 368-382, July, 1932, and Proceedings
of The Institute of Radio Engineers, Vol. 20, pp. 1117-1130, July, 1932.

6. U. S. Patent 1,565,548, December 15, 1925, issued to A. B. Clark.

7. U. S. Patent 1,737,830, December 3, 1929, issued to George Crisson,

8. U. S. Patent 1,738,000, December 3, 1929, issued to E. I. Green.

9. U. S. Patent 1,757,729, May 6, 1930, issued to R. C. Mathes.

The Efifect of Background Noise in Shared Channel
Broadcasting

By C. B. AIKEN

The interference which occurs in shared channel broadcasting consists
of several components of different types. Of these the program interference
is usually the most important in the absence of a noise background, while
if a strong noise background is present another component, which may be
called flutter interference, predominates.

A simple theory of the flutter effect is developed and it is shown that its
importance is dependent upon the type of detector employed. If manual
gain control is used, flutter may be greatly reduced by the use of a linear
rectifier. However, if automatic gain control is used this superiority of
the linear detector cannot be realized and flutter is bound to be troublesome.

The results of experimental studies of the various types of interference
are given and a comparison is made of the relative importance of flutter and
program interference. The effects of the type of detector used and of the
width of the received frequency band are observed. It is evident from
these studies that improvements in the size of the lower grade service areas
of shared channel stations might be obtained by close synchronization of
the carrier frequencies, even though different programs are transmitted.

THE regulation requiring that carrier frequencies be maintained to
within fifty cycles of their assigned values has resulted in the
practical disappearance from shared broadcast channels of the hetero-
dyne whistle, that most pernicious of all types of radio interference.
Consequently, it is now unnecessary to have so large a ratio of the
field strength of the desired signal to that of the undesired as was the
case before the banishment of the high pitched squeal. Nevertheless,
the field strength ratio which is necessary to permit of satisfactory
reception on shared channels is still much higher than we should like it
to be, and interference still abounds.

A very common type of interference is that which manifests itself
as a fluttering or heaving sound, often very unpleasant in character.
This phenomenon is caused by the periodic rise and fall of the back-
ground noise (static, R. F. tube and circuit noise, etc.) as the weak
interfering carrier wave swings alternately in and out of phase with
the carrier from the stronger station. In the complete absence of a
noise background, program interference, or "displaced sideband inter-
ference" ^ as it may be called, is more troublesome than are flutter
effects. Consequently, it is in regions other than the high grade
service areas of shared channel stations that flutter effects are most
annoying. In such regions they occur most prominently when the

^"The Detection of Two Modulated Waves Which Differ Slightly in Carrier
Frequency," Proc. I. R. E., January, 1931, and Bell. Sys. Tech Jour., January, 1931.

333

334 BELL SYSTEM TECHNICAL JOURNAL

frequency difference of the desired and interfering carriers is only a
few cycles per second. As this difference is increased the flutter is
transformed into a more sustained sound, rather harsh in character, and
as it is still further increased a low growl appears which becomes more
objectionable as it rises in frequency. The pitch of this growl cannot
exceed 100 cycles unless one or both stations are violating the 50 cycle
regulation. With the increasing use of very precise frequency control,
heterodyne frequencies of a few cycles have become very common, and
so, therefore, have flutter effects.

It has been pointed out in an earlier paper ^ that the magnitude of
the flutter effect will depend upon the type of rectifier employed in the
receiving set, and that it will be very much more objectionable when a
square law detector is used than when a linear detector is employed.
This is to be expected, since in the former case the audio-frequency
output of the receiver will be proportional to the amplitude of the in-
coming carrier, while in the latter case the output will be essentially
independent of the carrier amplitude, provided over-modulation does
not occur. However, these statements refer to the case in which
automatic gain control is not used. When the receiver is equipped
with automatic control, as in most better grade modern receivers, the
superiority of the linear detector is nullified and a serious flutter may
occur.

In addition to displaced sideband interference and flutter, trouble
may arise from distortion of the desired program by the action of the
interfering carrier. One or both of the first two types of interference
are likely to occur at lower field strength ratios than is the last, but at
higher levels of the undesired carrier all three types are of importance
and combine to degrade the quality of reception. In this paper,
studies of all these types will be reported. Audible beat interference
will not be discussed since it has been considered in other papers and,
as just mentioned, is much less important than it used to be.

Theoretical Estimation of Flutter Effects

As has already been stated, the flutter effect is due to the rise and
fall of the level of the noise background with variation in the effective
amplitude of the impressed carrier. In order to study this effect, let
us suppose that there are impressed upon the detector a component of
radio frequency noise which may be represented by iVcos (co -{- n)t, and
a desired carrier E cos w/. nj2-K is assumed to be an audio-frequency.

If a square law, or quadratic, detector is employed, the audio-

^" Theory of the Detection of Two Modulated Waves by a Linear Rectifier."
Proc. 1. R. E., Vol. 21, pp. 601-629, April, 1933.

BACKGROUND NOISE IN BROADCASTING 335

frequency output will be proportional to the audio-frequency compo-
nent of

[E cos ut -\- N cos (co + n)ty-,
which is

EN cos nt. (1)

Now suppose that there is impressed, in addition to the desired
carrier and noise component, a weak carrier e cos (w + u)t. The sum
of the strong and weak carriers may be conveniently regarded as a
single wave of amplitude

{E -\- e cos ut).

This may be substituted for the amplitude E in (1), giving for the
noise output

EN{1 -f K cos ut) cos nt (2)

in which

K = ejE. (3)

The noise which is heard will consist of a steady portion, the amplitude
of which is proportional to EN, and another portion of variable ampli-
tude which is proportional to ENK cos ut.

The factors that determine the importance of the flutter are many
and complex, but it seems likely that the most important of them is the
ratio of the variable component of the noise output to the steady com-
ponent. As long as the noise is loud enough to be obvious, this ratio
should be a fairly good measure of the perceptibility of the flutter,
and we shall venture to regard it as such. The experimental data to
be reported later will bear out this assumption.

From (2) it is evident that the ratio mentioned is merely K, the
ratio of the amplitude of the interfering carrier to that of the desired
carrier. We shall call this ratio the "flutter factor" for the quadratic
detector and designate it by Fq.

Fq^ K = e/E. (4)

It is interesting that Fq is independent of the amplitude N of the high
frequency noise.

It is possible to derive a similar factor, giving the ratio of the varia-
ble to the steady components of noise, for the linear detector. From
equations (70a) and (71) of the paper ^ already mentioned it follows
that the flutter factor for the linear detector, at low modulations of the

desired wave, is

Ne kK

^^^lE^^X' ^^^

in which k = N/E.

336 BELL SYSTEM TECHNICAL JOURNAL

Fl is seen to be dependent upon the strength of the high frequency
noise as well as upon that of the interfering carrier. It is also to be
noted that the flutter will be more serious with the quadratic than with
the linear detector by a factor 4/k = 4E/N, which is usually large.

This derivation of Fq and Fl on the basis of a single frequency
noise component serves to indicate important differences between the
two types of detector and to show how the flutter changes with the
noise level and with the ratio of the incoming carrier amplitudes. In
any practical case the noise field would consist of numerous frequency
components, but it is reasonable to expect that the proportionalities
expressed in (4) and (5) would still hold. However, the absolute
values of N and K at which the flutter becomes detectable must be
determined experimentally and may be expected to depend upon the
width of the received frequency band.

In the foregoing derivations it has been assumed that there is no
automatic volune control in the receiving set. A brief examination
of the effect of such a device will now be made.

Action of an Automatic Volume Control

The comparative freedom from flutter effects which has been noted
in the case of the linear detector may be regarded as due to the fact that
the audio-frequency output of such a detector is independent of carrier
amplitude over a wide range. If automatic volume control is used in
the receiving set, the amplitude of the carrier wave will be maintained
practically constant at the input terminals of the detector. If the
effective carrier amplitude impressed upon the antenna undergoes a
periodic fluctuation, due to very low frequency heterodyning between
the two stations, the gain of the radiofrequency amplifier will undergo
cyclic variations, so as to keep the carrier constant at the detector.
Obviously this will cause a fluctuation in the amplitude of the side-
bands, be they due to noise or program.

From this it is evident that, on the one hand, flutter effects in the
presence of a noise background will usually be of minor importance if a
good linear rectifier is employed in conjunction with a manual volume
control; while, on the other hand, these effects may become extremely
objectionable if automatic volume control is used. Because of the
prevalent use of AVC in modern radio receivers the low flutter char-
acteristics of the linear detector cannot be generally employed to
reduce flutter interference on shared channels.

In the case of the square law detector, the output is proportional
to the product of the amplitudes of the carrier and side frequencies.
At first glance it might seem that the use of automatic volume control

BACKGROUND NOISE IN BROADCASTING 337

should reduce the flutter effects, since it would iron out the variations
in carrier amplitude impressed upon the detector. However, it is
evident that this stabilization of the carrier will be exactly offset by
the variation imposed upon the sideband amplitudes, and that conse-
quently the flutter effects should be as evident when a normally func-
tioning automatic volume control is used as they are in the case of
manual control.

A perfectly functioning automatic volume control should make
flutter effects approximately independent of the type of detector em-
ployed when the beat frequency is of the order of 2 or 3 cycles. How-
ever, at some of the higher frequencies, of the order of 20 to 40 cycles,
the control will function with reduced efficiency, and at still higher
frequencies will not function at all. Consequently, in this intermediate
range the gain control may have some special effect and may make the
flutter either worse or better than it would be with the same type of
detector and manual control.

Experimental Studies
Equipment Used in the Study of the Effects of a Noise Background

A laboratory investigation was made of the interference between
two waves of slightly different carrier frequency. A block schematic
of the equipment used is shown in Fig. 1.

A modulated signal could be received from Station WABC, or, by
throwing the switch S, it was possible to obtain an unmodulated carrier
from a Western Electric No. 700A Oscillator, which is of very great
frequency stability.^ Whichever signal was used was fed through
an impedance matching transformer to a radio frequency attenuator.
The output of this attenuator was fed into the grid of one tube of a
mixing amplifier. As indicated in the drawing, this amplifier consists
merely of two shield grid tubes having a broadly tuned common plate
circuit load.

The other tube of the mixing amplifier was energized, through a
second radio frequency attenuator, by an unmodulated carrier derived
from a crystal controlled laboratory oscillator of the same type as that
which served as an alternative to WABC. This oscillator was part of
a Western Electric No. 1 A Frequency Monitoring Unit.^ The monitor
includes arrangements for measuring frequency differences between the
oscillator included within it and an external source. In this case the
external source was WABC, or the alternative carrier. The energy

3 0. M. Hovgaard, "A New Oscillator for Broadcast Frequencies," Bell Labora-
tories Record, 10, 106-110, December, 1931.

■• R. E. Coram, "A Frequency Monitoring Unit for Broadcast Stations," Bell
Laboratories Record, 11, 113-116, December, 1932.

338

BELL SYSTEM TECHNICAL JOURNAL

required by the frequency measuring device was supplied through a
tuned buffer amplifier.

The voltage developed across the tuned circuit of the mixing am-
plifier was measured by a conventional form of vacuum tube voltmeter.
By setting one attenuator at a very high loss, the magnitude of the sig-
nal supplied through the other could be measured, and the process then
reversed. If the two signals were adjusted so as to give equal ampli-
tudes across the tuned load, then any desired carrier ratio could be
obtained by adding a known loss in one attenuator.

NO.700A
OSCILLATOR

TUNED

RADIO
FREQUENCY
AMPLIFIER

VACUUM TUBE
VOLTMETER

RADIO'

FREQUENCY

ATTENUATORS

RADIO RECEIVER
.(SQUARE LAW
/i DETECTOR)

RADIO RECEIVER
(LINEAR DETECTOR)

NO.IA
FREQUENCY
MONITORING

UNIT

OS£i)

RADIO
FREQUENCY.
AMPLIFIERS

LOUD
SPEAKER

RADIO
FREQUENCY
ATTENUATOR

AUDIO
FREQUENCY

AMPLIFIER

VOLUME
INDICATOR

Fig. 1 — Schematic circuits of experimental setup.

The mixing amplifier fed a shielded transmission line which included
an adjustable pad. The line supplied energy to either of two radio
receivers, one of which contained a square law and the other a linear
detector. The output of the receiver was monitored on a loud speaker
and also on a volume indicator. Meters were provided for indicating
the change in direct current flow in the detector circuit of both receivers.

In order to study the effects of a noise background, a noise source
of constant and controllable level was required. Furthermore, it was
desirable that the noise be of a type frequently encountered in practice.
The thermal noise generated in a high gain amplifier seemed to be
suitable. Consequently, there were connected in cascade two ampli-
fiers having a gain of approximately 44 db each, over the entire broad-

BACKGROUND NOISE IN BROADCASTING

339

cast band. The output of the second of the units was fed through a
radio frequency attenuator to the grid of a single stage amplifier, the
output circuit of which contained a step-down transformer bridged
across the transmission line feeding the radio receivers. With zero
loss in the attenuator the noise energy fed to the line was ample for the
purposes of the present study.

An additional description of some of the pieces of equipment used
in the foregoing set-up may be of interest.

Source of Constant Unmodulated Carrier Frequency
The oscillator contained in the No. lA Frequency Monitoring Unit
is of unusual frequency stability. The piezo-electric crystal is mounted
in a specially designed thermal insulating chamber which reduces the
temperature fluctuations to an extremely small fraction of a degree.
Voltage regulating equipment is included in the unit, giving further
assistance in stabilizing the frequency. Detailed descriptions of the
oscillator ^ and of the frequency monitor ^ have been published.

A similar oscillator is used as a control unit at Station WABC.
Hence, it was expected that a very constant beat frequency could be
obtained between that station and the local oscillator. The frequency
of the latter was adjustable over a narrow range by means of a vernier
condenser in the crystal circuit. Figure 2 shows a number of plots of

^.-^

• ,

■ '

-•^

UJQ ^

Oz 0

(E UJ
UJoi

— .

fr^o

■" '^

5,-1 0

J-

""^

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

TIME IN MINUTES

Fig. 2 — Beat frequency between WABC and Western Electric No. lA Frequency

Monitoring Unit.

the beat frequency against time. These curves indicate an extremely
slow drift, and experience has shown that the beat frequency would
hold to within 0.4 cycle over a period of at least five minutes, and
usually considerably longer. This high stability greatly facilitated
work which required a very small difference in frequency of the two
carriers.

340

BELL SYSTEM TECHNICAL JOURNAL

Radio Receivers
Both receivers were high fidehty (7000 cycles) units of the tuned
radio frequency type. One of these was modified so that either man-
ual or automatic volume control could be used, and the level impressed
upon the detector was reduced so that it would function as a strictly
square law device. The cathode resistor which normally furnishes
a grid bias for the detector tube was replaced by a battery. This was
necessary in order to prevent straightening out of the characteristic
by degeneration at very low frequencies. In Fig. 3 is shown a plot of

4 00
300

ui 200

LU
Q.

g 100

(

S 50

^ 40

^ 30

LU
<

cc
o

UJ 10
Q

5 4

2
1

5 10 20 30 40 50 100 200 300 500

IMPRESSED ALTERNATING VOLTAGE IN ARBITRARY UNITS

Fig. 3 — -Characteristic of radio frequency amplifier and square law detector.

the change in detector space current as a function of the impressed
voltage. It will be observed that for increments of less than 200 ^A
the characteristic has a slope of two to one. All observations were

BACKGROUND NOISE IN BROADCASTING

341

made at signal levels which were low enough to stay well within this
range.

The other set was provided with a diode rectifier which functioned
as a linear detector. In order to improve the linearity of the charac-
teristic, an initial bias was used and was adjusted to obtain the best
characteristic as indicated by the following test:

If a large unmodulated carrier is impressed on a linear rectifier,
together with a much smaller unmodulated carrier, the beat frequency
output should be independent of the amplitude of the larger carrier
over a wide range. This phenomenon was observed experimentally
and the initial bias was altered until the range, over which the large
carrier could be adjusted without changing the output, was a maxi-
mum. In Fig. 4, the horizontal curve shows the magnitude of the

y 80

"^X

3 5

2 o;

20 30 40 50 60 70 80 90 100 110

IMPRESSED ALTERNATING VOLTAGE IN ARBITRARY UNITS

Fig. 4 — Characteristics of radio frequency amplifier and linear detector.

audio-frequency output, while the sloping curve shows the direct
current flowing in the detector circuit. The dashed curve is due to the

342 BELL SYSTEM TECHNICAL JOURNAL

presence of the weak signal, while the solid one represents the effect of
the large signal alone. The curves of this figure were taken with a
bias of + 0.5 volt, which was found not to be critical.

The results of the experimental observations made with this de-
tector were entirely in accord with theory, as will be discussed later,
while similar observations made with a zero bias gave results which
differed considerably from those predicted by the theory of the linear
rectifier. Lack of the small bias caused a considerable departure from
linearity, as was plainly evidenced by the fact that when it was absent
the audio-frequency output due to the two carriers was by no means
independent of the magnitude of the larger.

The tuned circuit in the mixing amplifier was so broad as to have an
entirely negligible effect on the fidelity of the radio receivers.

Listening Conditions

In studying the effects of noise background some observations were
made in the open laboratory, and a greater number in a partially
deadened room 10 feet x 10 feet x 10 feet. The sound-proofing of
this room was sufficient to keep out street noises and other extraneous
disturbances of moderate intensity.

In determining the dependence of a given effect upon the magnitude
of the carrier ratio, there was recorded that value of the ratio at which
the effect was just perceptible.

Results of Experimental Work

A number of observations have been made with the intention of
obtaining practical data on the characteristics of reception in the
presence of a noise background, and with the purpose of checking the
theoretical predictions already given. It has been pointed out that the
flutter effects depend upon the type of detector which is employed and
upon the ratio of the two carriers. If a square law detector is used the
effect should be very nearly independent of the magnitude of the noise
level, so long as it is within reasonable limits and does not either over-
load any of the equipment (including the ear of the listener) or fall so
low as to be hardly noticeable. On the other hand, if a linear detector
is employed, flutter effects should increase with the noise level. In
either case the modulations of the two stations play no important part
in determining the flutter effects except in so far as high modulations
may temporarily mask them.

As a result of these considerations it was decided to employ un-
modulated carriers for the greater part of the work. In order that a
suitable level might be chosen, the strong carrier was first adjusted to

BACKGROUND NOISE IN BROADCASTING

343

give the proper change in detector current. It was then modulated
30 per cent with a pure tone, and the gain of the audio-frequency out-
put ampHfier was adjusted until a fairly loud, but entirely comfortable,
level was delivered to an observer placed about six feet in front of the
loud speaker. The output level of the audio-frequency amplifier was
read on a meter so that its gain might be checked later on.

The Linear Rectifier
The detector of a radio receiver was adjusted to have a linear recti-
fier characteristic in the manner just described and manual gain con-
trol was employed. In the first set of runs the carrier ratio was de-
termined at which the flutter effect at low frequencies, or the carrier
beat-note at higher frequencies, became just noticeable, the frequency
being the variable. In Fig. 5 is shown a curve representing a number of

. —^-

< 10

0 10 20 30 40 50 60 70 80 90 100 110

BEAT FREQUENCY IN CYCLES PER SECOND

Fig. 5 — Carrier ratio for perceptible flutter with a linear detector. Noise equivalent
to 9.5 per cent modulation.

observations of this type. The noise level was constant at 10 db
down from a 30 per cent modulated signal. By this it is meant that
when the noise was impressed upon the receiver, together with a car-
rier the level of which had been fixed as described above, the audio-
frequency output, as measured on a copper oxide level indicator, was
10 db below the audio output resulting from a 30 per cent modulation
of the same carrier in the absence of noise.

A very interesting fact to be noted from this curve is that, for beat
frequencies of less than about 20 cycles, the carriers must be very nearly
equal before any flutter effect whatever may be detected. The
average curve has been drawn through a value of 1.5 db. The ob-
served values vary from this figure by not more than ±0.5 db.

344

BELL SYSTEM TECHNICAL JOURNAL

The right-hand portion of the curve is determined by the audibility
of the beat-note, and its position will of course depend upon the masking
effect of the noise background. Theory has indicated that the flutter
frequency portion of the curve should drop with the noise level, but it
is evident that, with such a small difference in carrier amplitudes as
is indicated in the figure, the results would not be appreciably differ-
ent were the noise level to be reduced. On the other hand a noise
level which is down only 10 db from a 30 per cent modulated signal is
equivalent to a modulation of nearly 10 per cent. This is an extremely
objectionable noise level, so objectionable, in fact, that under the condi-
tions of the tests it was very unpleasant to listen to. Consequently,
it did not seem worth while to run curves similar to that of Fig. 5 for a
number of different noise levels. Instead, a set of observations was
made with a fixed carrier frequency difference of 2 cycles and a variable
noise level. The results are indicated by the lower curve in Fig. 6.

f ■ — ■ —

<; (»

> I 1 1 1 1 1 1 1 1 1 1

0 5 10 15 20 26 30

NOISE LEVEL IN DECIBELS BELOW 30 "/o EQUIVALENT MODULATION

Fig. 6 — Carrier ratio for perceptible flutter as a function of noise level. The upper
curve is for the square law and the lower curve for the linear detector.

With a noise level equivalent to a 30 per cent modulation, a carrier ratio
of only 2 : 1 is necessary to reduce the flutter to a barely detectable
amount. At low noise levels, down 20 db or more from 30 per cent, the
flutter could hardly be detected but there was noticeable a "bumping"
sound which was due to the rather violent motion of the cone of the
loud speaker at a frequency of 2 cycles. This was partially eliminated

BACKGROUND NOISE IN BROADCASTING

345

by inserting a capacity in series with the voice frequency circuit of the
speaker, but even when greatly reduced the bumping was detectable
and was more important than any flutter which may have been present.

The Square Law Rectifier
Observations similar to those just discussed were made with a square
law detector. In Fig. 7, the ordinates represent the carrier ratio neces-

— — ^ — . — 1

30 40 50 60 70 80
BEAT FREQUENCY IN CYCLES PER SECOND

Fig. 7 — Carrier ratio for perceptible flutter with a square law detector. Noise
equivalent to 9.5 per cent modulation.

sary to reduce the flutter to a just detectable value, while the abscissae
represent the beat frequency. The noise is 10 db down from an equiva-
lent 30 per cent modulation. The curve is in striking contrast to that
of Fig. 5. At very low frequencies a carrier ratio of 28 db is required
when a square law detector is employed, while if the receiving set
embodies a linear detector a ratio of 1.5 db is sufficient. The right-
hand portions of the curves are fairly similar, since the carrier ratio is
here dependent upon the audibility of the beat note and not upon
flutter effects. The observations of which Fig. 7 is a record were made
in the small sound-proof room. In Fig. 8 are shown two curves made
in the open laboratory. In the upper curve the noise output was ap-
proximately 20 db down from that due to a 30 per cent modulated
signal, while in the lower curve it was approximately 30 db down.

The theory which has been outlined indicates that in the case of the
square law detector the flutter eff^ects should be practically independent
of noise level, and the curves shown in the last three figures bear out
this prediction quite positively. Even more definite confirmation is
furnished by the upper curve of Fig. 6, which shows the result of ob-
servations taken with a fixed beat frequency of 3 cycles. The two
curves of this figure show the great superiority of the linear rectifier
over the square law in receiving non-isochronous transmissions in the
presence of a noise background.

346

BELL SYSTEM TECHNICAL JOURNAL

Z 12

t 28

••

•X

^ —

12 16 20 24 28 32 36

BEAT FREQUENCY IN CYCLES PER SECOND

Fig. 8 — Carrier ratio for perceptible flutter with a square law detector. Noise
equivalent to 3 per cent modulation, for the upper curve, and to 0.95 per cent for the
lower curve.

The Square Law Rectifier with Automatic Volume Control

It has been predicted that the use of automatic volume control in
the receiving set should greatly increase the flutter effects observable
with a linear rectifier, while with a square law device these effects
should be the same for both automatic and manual control except,
perhaps, at the frequencies of reduced efficiency of the gain control.
An experimental check was made on the latter statement, the results
of which are shown in Fig. 9. It will be noticed that this curve is very
similar to the curves of Figs. 7 and 8.

,20

—'

••• ,

•

30 40 50 60 70 80

BEAT FREQUENCY IN CYCLES PER SECOND

Fig. 9 — Carrier ratio for perceptible flutter with automatic volume control. Noise
equivalent to 9.5 per cent modulation.

BACKGROUND NOISE IN BROADCASTING 347

The action of an automatic volume control, in keeping constant the
level of the total carrier delivered to the detector, should become less
pronounced as the beat frequency rises and should fail altogether when
this frequency reaches the audible range. This reduction in efiticiency
of control may either increase, leave unaltered, or decrease the magni-
tude of the flutter, depending upon the amount of time delay involved
in feeding back the controlling voltage. In the receiver used, the re-
duction in efficiency of the gain control occurred between 20 and 40
cycles. A comparison of Fig. 9 with Figs. 7 and 8 indicates that in this
receiver the gain control tends to increase the flutter somewhat when
the heterodyne frequency is within this range.

Interference of Undesired Program

When the interfering station transmits a program which is different
from that of the desired station, serious interference may occur which
is due primarily to the beats between the undesired sidebands and the
desired carrier. If the carrier beat frequency is subaudible and there
is little or no noise background, this will be the predominant form of
interference. Its magnitude will depend upon the degree of modula-
tion of the undesired signal, but is practically independent of the type
of detector and gain control which are used. In the presence of con-
siderable noise background it may or may not be more important than
flutter effect.

In order to get some data on this point, observations were made
with a square law detector and manual gain control. This represents
about the worst condition, as far as flutter effect goes, but will be ap-
proximated by AVC receivers. At a fixed noise level the carrier ratio
was determined at which the flutter could be noted, and also the ratio
at which the program interference was detectable. This was done for
receiver band widths of 7000 and 3500 cycles. The band width had
no appreciable effect upon the program interference but exercised a
very definite effect upon the flutter. Fig. 10 shows the results of the
observations which were taken. The solid sloping curve represents
the average of the observations on program interference, while the two
horizontal curves show the carrier ratio at which the flutter was just
detectable for the two bands widths used. The program interference
was classed as audible when it could just be heard on the peaks of modu-
lation. However, for considerable intervals of time it was entirely
inaudible. Consequently, when the same carrier ratio was recorded
for the flutter and for the program interference the former was actually
the more annoying. In order to take account of this difference of
character between the two types of interference it is necessary to

348

BELL SYSTEM TECHNICAL JOURNAL

shift the program curve downward. Just how far it should be dis-
placed is very hard to determine, as the amount will depend upon the
type of program on the undesired station. Observations have indi-
cated that the shift should amount to at least 7 db. The dashed curve
in Fig. 10 has been drawn 7 db below the solid curve.

-I 30
lij
m

^20

•

PROGRAM INTERFERENCE

^^--"^ 1

A FLUTTER INTERFERENCE, 3.5 KC BAND

1 1
-I''

. ' '

.^---\

^ 1

^"^

.^^

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

NOISE LEVEL IN DECIBELS BELOW 30% EQUIVALENT MODULATION

Fig. 10 — Relative importance of program interference and flutter interference.

It will be noticed that with a band width of 3500 cycles the flutter
curve crosses the program curve at a noise level equivalent to about
2 per cent modulation. (In every case the noise level was measured
with the 7000 cycle band, regardless of what band was to be used in the
listening tests. This should be kept in mind throughout the present
discussion.) This means that at equivalent modulations of more than
2 per cent the flutter effect would be more objectionable than the
program interference. However, at high noise levels, say 5 to 10 per
cent, the listener would be sure to reduce the band width of his receiver
to considerably less than 3500 cycles and this would reduce the relative
importance of the flutter. Nevertheless, at very high noise levels the
flutter is more important than the program interference. If the un-
desired station were to employ abnormally low modulation the program
interference would be decreased and the relative importance of the
flutter increased.

It is evident that the dependence of the flutter on band width, and
the different reaction of individual observers as to what type of inter-
ference is the more objectionable, renders it impossible to make a definite
statement as to the exact values of carrier ratio and noise field which
will make the two types of interference equally important. But we
can draw the useful conclusion that in cases of excessive noise, such

BACKGROUND NOISE IN BROADCASTING 349

as may occur in rural areas without causing the Hstener to abandon
attempts at reception, the flutter will be the more important. Conse-
quently, an improvement in the service in such regions would be ob-
tained by synchronizing the carriers of the two stations, even though
they continue to transmit different programs.

Effect of Interfering Carrier on Desired Program

Even if the interfering wave were unmodulated and there were a
neglibible noise background, there still remains the possibility of dis-
tortion of the desired program by the heterodyning action of the un-
desired carrier. In order to determine how important this effect is as
compared with those which have been discussed, a modulated carrier
(derived from WABC) and a w^eaker unmodulated wave were used.
A beat frequency of about 3 cycles was maintained during the course
of these observations.

With the linear rectifier it was found that a perceptible distortion
of the desired program could not be detected on speech and jazz music
until the weak carrier was brought within 1 db of the strong one.
When the program consisted of music containing many sustained notes,
such as occur in a violin solo and even in vocal solos, the cyclic varia-
tions in output level were more noticeable. In such a case a ratio
of about 4 db was necessary to reduce the distortion to the detectable
limit.

With the square law rectifier it was found that a carrier ratio of
10 db produced detectable distortion with any type of program. At a
ratio of 16 db distortion could be detected only when the program con-
tained sustained notes, and at 18 db could be noticed only when the
notes were sustained for a considerable time.

The dependence of the permissible ratio upon the type of program
led us to make a similar observation when the strong carrier was modu-
lated 30 per cent with a pure tone of 400 cycles. Under such condi-
tions it was necessary to reduce the interfering carrier to about 34 db
below the strong one before the 3-cycle variation in the pure tone
definitely vanished.

Conclusions

The studies which have been reported furnish quantitative data on
the various types of interference which are encountered in shared
channel broadcasting and show what relative levels of interfering
carrier may be tolerated under various conditions.

In high grade service areas the program from the undesired station
will be the most serious form of interference, provided the carrier beat

350 BELL SYSTEM TECHNICAL JOURNAL

frequency is subaudible. If there is a moderate noise background
present, it will tend to mask the program and will therefore permit
of somewhat higher interfering field strength. However, if the inter-
ference is raised beyond a certain level, dependent upon the received
band width, flutter effects will become pronounced. This will not be
true with a linear detector and manual gain control, but in practice
radio receivers which have linear detectors almost invariably have
automatic volume control.

If the noise level is very high it may mask even rather loud program
interference, and under such conditions the flutter effect is likely to be
much the most serious source of trouble. This condition is of practical
occurrence in outlying areas where a degraded service must be toler-
ated continually. In such regions shared channel broadcasting is
limited in usefulness primarily by the flutter effects, and in extreme
cases, by distortion of the desired program due to the heterodyning
action of the interfering carrier. Both of these types of disturbance
would be eliminated by synchronizing the carriers of the two stations,
and it seems likely that control of the carrier frequencies to within
±0.1 cycle might definitely extend the limits of the lower grade service
areas of shared channel stations.

Acknowledgment

I wish to acknowledge my indebtedness to Mr. J. E. Corbin for his
assistance in carrying out the experimental work which has been
reported in this paper.

Wide-Band Open- Wire Program System *

By H. S. HAMILTON

Radio programs are regularly transmitted between broadcasting stations
over wire line facilities furnished by the Bell System. Both cable and
open wire facilities are employed for this service. Recently a new program
transmission system for use on open wire lines has been developed which
has highly satisfactory characteristics. A description of this open wire
system and test results obtained with it are given in this paper.

THE simultaneous broadcasting of the same radio program from
a large number of broadcasting stations, in different sections of
the United States, has become of such everyday occurrence that the
radio listening public takes it as an accepted fact and in many cases
does not know whether the program is originating in the studio of a
local broadcasting station or in a broadcasting studio in some distant
city. The wire line facilities furnished by the Bell System for the
interconnection of the radio stations, particularly the wire line facilities
in cable, have such transmission characteristics that little detectable
quality impairment is introduced even when programs are transmitted
over very long distances.

This cable program system was described in a recent paper.^ More
recently a new program system for use on open-wire lines, which
possesses transmission characteristics comparable with those of the
cable system, was developed and an extensive field trial made involving
two circuits between Chicago and San Francisco. This paper describes
this new open-wire program system and gives the principal results of
the tests made on the two transcontinental circuits.

In the paper referred to describing the cable system, the various
factors and considerations involved dictating the grade of transmission
performance that is desired for program circuits were discussed in
considerable detail so they will not be reviewed here. The transmis-
sion requirements chosen as objectives for both cable and open wire
are as follows:

Frequency Range

Frequency range to be transmitted without material distortion —
about 50 to 8,000 cycles.

* Published in April, 1934 issue of Electrical Engineering. Scheduled for presen-
tation at Pacific Coast Convention of A. I. E. E., Salt Lake City, Utah, September,
1934.

1 A. B. Clark and C. W. Green, " Long Distance Cable Circuit for Program Trans-
mission," presented at A. L E. E. Convention, Toronto, June, 1930; published in
Bell Sys. Tech. Jour., July, 1930.

351

352 BELL SYSTEM TECHNICAL JOURNAL

Volume Range

Volume range to be transmitted without distortion or material
interference from extraneous line noise — about 40 db which
corresponds to an energy range of 10,000 to 1.

Non-Linear and Phase Distortion

Non-linear distortion with different current strengths and phase
distortion to be kept at such low values as to have negligible
effect on quality of transmission even on the very long circuits.

The frequency range afforded by the new open-wire program circuits
extends about 3,000 cycles higher and more than 50 cycles lower than
the frequency range available with the open-wire ^ program circuits
previously used. The extension of the frequency range at the upper
end necessitates the sacrifice of one carrier telephone channel of carrier
systems operating on the same wires with the program pair since the
frequency band of the lowest carrier channel lies in this range. In
order to minimize noise and the possibility of crosstalk, the phantoms
of program pairs are not utilized and, of course, d.-c. telegraph com-
positing equipment is removed in order that the proper low-frequency
characteristics may be realized.

Description of New Open-Wire System
In general, the amplifiers on the open-wire program circuits employ
the same spacing as the telephone message circuit repeaters on the
same pole lead. The average repeater spacing is about 150 miles
but the repeaters may be located as close as 60 miles or may be as
much, as 300 miles apart depending on the location of towns and cities
on the open-wire route and the gauge of the wires used. The upper
diagram of Fig. 1 shows a typical layout of the new wide-band open-
wire program system. Three types of stations are shown, a terminal
transmitting station, an intermediate station which may be either
bridging or non-bridging and a terminal receiving station.

The terminal transmitting station includes an equalizer for correct-
ing for the attenuation distortion of the local loop from the broad-
casting studio, an attenuator for adjusting the transmission level
received from the local loop to the proper value, an amplifier for in-
serting the required gain, filters for separating the program and carrier
channels, monitoring amplifier, loudspeaker and volume indicator for

^ A. B. Clark, "Wire Line Systems for National Broadcasting," presented before
the World P'ngineering Congress at Tokio, Japan, October, 1929; published in Proc.
L R. E., November, 1929, and in Bell Sys. Tech.^ Jour., January, 1930. F. A.
Cowan, "Telephone Circuits for Program Transmission," presented at Regional
Meeting of A. I. E. E., Dallas, Texas, May, 1929; published in Transactions of
A. L E. E., Vol. 48, No. 3, pages 1045-1049, July, 1929.

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

353

5°

3°

s ^

E ^

Hfl EEH [U 5

3 ? 3

^] [^ A(?)Et] ^

< —

° 2

< _,

C3l:^^Cil>^

RELATIVE VOLUME
IN DECIBELS

354 BELL SYSTEM TECHNICAL JOURNAL

making the necessary operating observations and a predistorting net-
work and associated amplifier.

At the intermediate station are included line filters for separating
the carrier currents and program currents and directing them to their
proper channels, two adjustable attenuation equalizers for correcting
for the attenuation distortion of the line wires and associated appa-
ratus, gain control attenuator, line amplifier and associated monitoring
equipment. At intermediate stations where it is necessary to provide
branches to radio stations or to other program circuits an amplifier of
a special type having several outlets is inserted immediately in front
of the line amplifier.

At a receiving terminal, the layout employed is very similar to that
utilized at intermediate stations except that an additional low-pass
filter and a restoring network are inserted ahead of the receiving
amplifier.

A novel feature is provided in this program system for minimizing
its susceptibility to interference at higher frequencies. It consists in
predistorting the transmission at the sending end of the circuit so
that currents above 1,000 cycles are sent over the line at a higher level
than if this arrangement were not employed, thus increasing the signal-
to-noise ratio at these frequencies. Such an increase in power at high
frequencies is permissible without overloading in the line amplifiers
in view of the fact that the energy content of the program material
above 1,000 cycles is materially less than at the low frequencies and
decreases rapidly as the frequency is increased. In order to restore
the program material to the same relations it would have if it were not
predistorted, a network is inserted at each point in the branches
which feed the radio stations and at the receiving terminal. This
network introduces attenuation and phase distortion which are com-
plementary to those introduced at the sending end of the circuit by
the predistorting network. The net reduction in high-frequency inter-
ference is equal to the discrimination introduced by the predistorting
network in favor of these frequencies, and is therefore equal approxi-
mately to the loss of the restoring network at the same frequencies.

In the lower part of Fig. 1 is shown a level diagram, from which
may be noted the losses and gains introduced by different parts of the
system at a frequency of 1,000 cycles. The maximum volumes which
are permitted in the various parts of the system are also indicated
approximately by this diagram.

Line Facilities
As is well known the open-wire lines employed in telephone and
program service do not have uniform attenuation for all frequencies,

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

355

the low frequencies being transmitted with much less loss than the
high frequencies. Since the program circuits employ the same type
of open-wire facilities that is used in the message circuits, three
different gauges of wire with either of two pin spacings between
wires may be used and the repeater sections may vary in length from
60 to 300 miles. This means that the attenuation frequency char-
acteristic of a repeater section not only varies with frequency but also
varies considerably in magnitude of attenuation depending on gauge
of wire and length of repeater section.

On Fig. 2 are shown three pairs of characteristics which illustrate
the loss-frequency characteristics of three lengths of 165-mil, 8-inch

20
18
16
14
12
10
8
6

■■

/
/

WET

DRY

/•

•

300

•

— —

—

'Eob

_miles_^

■ '

--_

,*^^-

__^

■-'

•■

—

2
0

—

100 500 1000

FREQUENCY IN CYCLES PER SECOND

10,000

Fig. 2 — Loss of 165-mil. 8-inch spaced pairs when inserted between 600-ohm

resistances.

spaced circuits. The lengths chosen for purposes of illustration are
100, 200 and 300 miles, respectively. The solid line curves show
the insertion loss-frequency characteristics of the circuits for average
dry weather conditions when the circuits are connected between 600-
ohm resistances. The dashed line curves indicate the wet weather
insertion loss characteristics, that is, they indicate the loss-frequency
characteristic which might obtain if the lines were very wet for the
entire length of a repeater section.

For the purpose of comparing the attenuation frequency character-
istics of the different types of open-wire lines, the curves shown on Fig.
3 have been prepared. These characteristics have been plotted so

356

BELL SYSTEM TECHNICAL JOURNAL

■

DRY WEATHER

100 MILES 165 MIL

/ >

I75MILES I28MIL

200 MILES 165 MIL

300 MILES 165 MIL

'/'/

,--'

— ';-

^.'

._.

;;

— '

—

'•^

20 50 100 500 1000 5000 10,000

FREQUENCY IN CYCLES PER SECOND

Fig. 3 — Attenuation characteristics of 8-inch spaced open-wire pairs when inserted
between 600-ohm resistances.

that all coincide at 1,000 cycles; thus a direct comparison of the differ-
ence in shape of the attenuation frequency characteristics may readily
be observed.

Figure 4 shows resistance and reactance components of 165-mil
and 128-mil 8-inch spaced open-wire lines. Note that, except at low
frequencies, the impedances of the various open-wire lines are quite
uniform throughout the frequency range and do not depart greatly
from 600 ohms. For this reason and in consideration that the majority
of telephone apparatus is designed for 600-ohm impedance, all units of
this new program system, except the carrier line filters, have been
designed to have an impedance of 600 ohms. In order to reduce
reflection losses, particularly in the carrier range, the line filters have
been designed to have an impedance on the line side somewhat lower
than 600 ohms although the drop or office side impedance is 600 ohms.

Attenuation Equalizers
To furnish the necessary attenuation corrections for the three
different gauges of lines, four adjustable attenuation correcting net-
works have been provided. One attenuation equalizer provides atten-
uation correction for high frequencies only and is common for all
gauges. The three other equalizers provide low-frequency attenuation
correction designed specifically for the particular gauge of circuit the

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

357

Z 900

< 800

£ 700

500

6 -100

-400

^128 MIL
^165

\ 165 MIL
^128

12 3 4 5

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 4 — Impedance'of 8-inch spaced open- wire pairs.

^ J. S—l "1 JOUT j,_ SIR, I

s—i,

. . n~l.fr

JJ [IL JJ U JJlL

SECTION 2 SECTION t

Fig. 5 — Low-frequency attenuation equalizer

BASIC SECTION

358

BELL SYSTEM TECHNICAL JOURNAL

equalizer is to be associated with and also include a fixed amount of
high-frequency attenuation correction.

On Fig. 5 is shown a schematic diagram of one of the low-frequency
attenuation equalizers. This consists of four sections of 600-ohm
constant impedance type networks. One section referred to as a basic
section introduces attenuation correction over the complete frequency
range from 35 to 8,000 cycles for a particular minimum length of line,
as for example, in the case of 165-mil circuits this is for 100 miles.
The three other sections on the other hand furnish attenuation correc-
tion only for frequencies from approximately 1,000 cycles down to
35 cycles. Section 1 of the equalizer for 165-mil circuits puts in about
}/2 db more loss at low frequencies than it does at 1,000 cycles. Sec-
tion 2 puts in double the amount of correction that is introduced by
Section 1 and Section 4 introduces four times as much attenuation
correction as Section 1. These three sections are controlled by
switches so that any one or all of them may be cut in tandem with the
basic section. The attenuation corrections afforded for the various
adjustments of this equalizer are shown on Fig. 6.

Q 4

is.

—

■^

—

B+?
B-f-'

*■/

!^^^^^

—

■— -~

f ■

BASIC
SECTION
ALONE

30 50 100 500 1000 5000 10,000

FREQUENCY IN CYCLES PER SECOND

Fig. 6 — Attenuation correction furnished by low-frequency equalizer for 165-niil.

circuits.

The attenuation equalizers for 128-mil and 104-mil facilities are
similar in construction to the one just described having different
constants so as to furnish somewhat different attenuation correcting
characteristics.

Figure 7 shows a schematic diagram of the high-frequency attenua-
tion equalizer. This consists of four 600-ohm constant impedance
type network sections which, as indicated, are controlled by switches
so that any one or all of them may be cut in tandem with the program

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

359

i 6 I I [OUT I jryi I I s~;ii I out| I s — i I

9 9 '^V V ' 9 <? 9

SECTION 8

SECTION 4

SECTION 2

SECTION 1

Fig. 7 — High-frequency attenuation equalizer.

circuit as required. Figure 8 shows the loss-frequency characteristics
of these four sections. As may be noted, the loss of the various
sections is practically constant over the frequency range up to 1,000
cycles, decreasing from there on to a minimum value at 8,000 cycles.

aj 3

SECTION 8

_4_

"^''''v

30 50 100 500 1000 5000 10,000

FREQUENCY IN CYCLES PER SECOND

Fig. 8 — Attenuation correction furnished by high-frequency equalizer.

Section 1, as may be noted, furnishes about 3^ db attenuation correc-
tion. Section 2 is double that of Section 1, Section 4 is four times that
of Section 1 and Section 8, eight times that of Section 1. These
sections may be used in tandem so that attenuation correction for the
high frequencies is, therefore, provided in steps of ^ db from zero to
71/2 db.

360

BELL SYSTEM TECHNICAL JOURNAL

An illustration of how the equalizers introduce the necessary attenua-
tion correction is given on Fig. 9. The lower curve on this figure shows
the loss of a 300-mile section of 165-mil circuit. The losses introduced
by the particular sections of low and high-frequency equalizers that

500 1000

FREQUENCY IN CYCLES PER SECOND

10,000

Fig. 9 — Loss of 300-mile line section and associated equalizers.

would be required for this length of line are indicated by the cross-
hatched areas, and the total line and equalizer loss is shown by the
top horizontal line. Sufficient gain is introduced by the line amplifier
to annul this loss.

Amplifiers

Two types of amplifiers are provided, one of which is used as a
line or monitoring amplifier and the other which is used as a means
for transforming one circuit into several circuits so as to feed various
branches at points required.

For certain combinations of program circuits as many as 50 ampli-
fiers may be connected in tandem. This necessarily imposes severe
requirements on the transmission performances of the amplifiers par-
ticularly with reference to flatness of gain-frequency characteristics
and phase distortion. By designing the coils used in the amplifiers

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM 361

so as to have very high inductances the desired phase distortion re-
quirements were met while at the same time the necessary flatness of
gain characteristic was obtained at the low frequencies.

Figure 10 shows the transmission circuit of the Una amplifier and
monitoring amplifier. This device has a 600-ohm input and output
impedance and consists of two stages of push-pull amplification. The
potentiometer is a balanced slide wire having a continuous gain adjust-
ment over a range of 6 db. A balanced input transformer serves to
connect the potentiometer to the grids of the two push-pull vacuum
tubes which function as the first stage of this amplifier. The first
stage is connected to the second or power stage by means of resistance
coupling which gives better results both as to phase distortion and
low-frequency gain characteristics than if transformer or retard coil
coupling were used. Resistances are provided in the grid circuits of
the second stage so that the high-frequency characteristic may be
adjusted as required. The power tubes are connected to an output
transformer which has the unique feature of providing a monitoring
outlet which is not materially affected by voltages produced at or
beyond the line terminals. The transformer, as may be observed,
consists of three balanced windings arranged as in the form of the
well-known hybrid coil used in two-wire telephone repeaters, with the
exception that the two low impedance windings are of unequal ratio,*
the line windings having many more turns than the monitoring wind-
ings. The ratio of the windings is such that the voltage at the monitor-
ing terminals when said terminals are closed through 600 ohms is 30
db below the voltage at the line terminals. Resistances are inserted in
series with the monitoring winding so that an impedance of 600 ohms
will be presented at the monitoring terminals.

The average gain of the amplifier with the potentiometer set at its
maximum position is 33 db. Of 100 amplifiers measured, the gain
at 35 cycles averaged .10 db less than the gain at 1,000 cycles while
from 100 to 8,000 cycles the gain was constant within .05 db. The
delay at 50 cycles is approximately .6 millisecond greater than it is
at 1,000 cycles. From 150 to 8,000 cycles the delay is substantially
constant and is only a small fraction of a millisecond. The amplifier is
capable of handling an output power 9 db above reference volume
without noticeable distortion.

At several points along a program circuit taps or branches are pro-
vided so as to connect various broadcasting stations to the program
circuit and also to connect to other program circuits which form part
of a broadcasting network. Points where such connections or branches
are made are commonly called bridging stations. At some points as

362

BELL SYSTEM TECHNICAL JOURNAL

5100

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM 363

many as six branches are supplied but generally only two or three taps
are utilized.

To accomplish this branching out at a bridging station a resistance
network multiple is provided, having six outlets. This network
multiple is shown on Fig. 11. To annul the loss of the network a
single stage amplifier is connected in front of it. This network mul-
tiple and amplifier are mounted on the same panel forming a single
integral unit. The network multiple is so proportioned that if any
one of the branches is accidentally opened or short-circuited the other
branches are affected to only a minor degree. The amplifier is ad-
justed so that the gain from the input terminals to any of the output
branches is zero. The bridging amplifier is normally inserted imme-
diately in front of the line amplifier. As in the case of the line
amplifier mentioned above, high inductance coils are utilized in order
to keep phase distortion at a minimum. A resistance adjustment is
provided in the grid circuit in order to adjust the high-frequency
characteristic of this amplifier to the desired value.

The gain-frequency characteristic of the bridging amplifier is prac-
tically identical with the corresponding characteristic just described
for the line amplifier, while the delay is even less.

Predistortion

The means utilized to accomplish the predistorted transmission
referred to earlier includes the provision of a so-called predistorting
network at the sending end of a program circuit and a restoring
network in each branch which supplies a broadcasting station. The
predistorting network introduces a large loss at low frequencies with
a decrease in loss as the frequency is increased. By introducing
suitable amplification immediately behind the predistorting network
the resultant effect is to raise the high-frequency transmission relative
to the low-frequency transmission by the difference in loss between
the 1,000-cycle loss of the predistorting network and its higher fre-
quency loss. The restoring network characteristic is the inverse of
the predistorting network. These two networks are 600-ohm constant
impedance type structures. The restoring network is shown schemat-
ically in Fig. 12. The predistorting network is generally similar to
this, having different constants and a slightly different arrangement
of elements. On Fig. 13 are shown the loss-frequency characteristics
of the predistorting and restoring networks and a third characteristic
which is the sum of these two. As may be noted this latter character-
istic has a constant value throughout the frequency range.

364

BELL SYSTEM TECHNICAL JOURNAL

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

365

■A/W

Fig. 12 — Restoring network.

SUM

^.^

\^PREDISTORTING

.^^ESTORING

—

■

0 1000 2000 3000 4000 5000 6000 7000 80(
FREQUENCY IM CYCLES PER SECOND

Fig. 13 — Attenuation characteristics of predistorting and restoring networks.

Line Filters

As a rule, on open-wire circuits other transmission channels are pro-
vided on the same wires which carry the program transmission.
These other channels operate at frequencies above the program range
and in order to direct the various currents to their proper channels
at a terminal or repeater station carrier line filter sets are inserted at
the ends of the line wires. The carrier line filter sets include a low-
pass and a high-pass filter. The low-pass filter, cutting off somewhat
above 8,000 cycles, directs the program transmission to the program

366

BELL SYSTEM TECHNICAL JOURNAL

apparatus and the high-pass filter which has a low end cutofif around
9,000 cycles directs the carrier transmission to its associated carrier
equipment. Attenuation frequency characteristics of these filters are
shown on Fig. 14. The low pass filter is of unusual design and is
described at some length in a companion paper.*

O 50

^_^

LOSS FROM^
A TO C

[

— ■

-LOSS FROM
A TO B

FILTERS

HIGH-
. PASS

(C)TO CARRIER
EQUIPMENT

TO LINE®

- LOW-
. PASS

(B)TO PROGRAM
EQUIPMENT

8 10 IE 14 16 18 20 22

FREQUENCY IN KILOCYCLES PER SECOND

24 26

Fig. 14 — Attenuation characteristics of line filter set.

Monitoring Features

A very important factor in the satisfactory operation of a program
system is the provision of monitoring arrangements by which the
operating forces are enabled to observe the quality of transmission,
listen for extraneous interferences and observe indicating devices in
order to make certain that the program is maintained at its proper
volume.

Three types of aural monitoring facilities were provided on a trial
basis for the new program system. The first type consists of a single
unit loudspeaker operated by a suitable amplifier. With this loud-
speaker system a good response characteristic from approximately 100
to 5,000 cycles is obtained, the low-frequency response depending, of
course, on the size of the baffle used with the loudspeaker.

The second type of monitoring consists of two headset receivers

arranged with a proper equalizing network circuit. This type of

monitoring provides good response characteristics from approxi-

3 A. W. Clement, "Line Filter for Wide-Band Open-Wire Program System,"
published in this issue of the Bell Sys. Tech. Jour.

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM 367

mately 50 cycles to 8,000 cycles, enabling the observer to cover the
entire program frequency range, thus permitting him to detect any
extraneous interference which may be introduced even though this
occurs at very low or very high frequencies.

The third type of monitoring consists of two loudspeakers and
associated equalizing network with the loudspeakers mounted in a
large bafifle board. This arrangement affords a fairly uniform response
from about 40 cycles to above 8,000 cycles. The particular type of
monitoring which might be provided at the various stations would
be governed by the service requirements involved.

To observe the volume on the program circuit, volume indicators
are used. A new type of volume indicator was made available along
with the new program system. This new device utilizes a full-wave
copper oxide rectifier, has a much greater sensitivity range than that
of the devices formerly used and possesses materially improved indicat-
ing characteristics. The volume indicator is connected across the
monitoring terminals of the line amplifier, in which position it is
bridged across a practically non-reactive 600-ohm impedance. Lo-
cated thus it is also independent of line impedance affording more
accurate results and obviating the necessity of correcting volume
readings on account of line impedances. Also at this location it
introduces no loss or phase distortion to the through program circuit.

The above constitutes a description of the major items employed
in this program system. There are a number of other units, such as
attenuators, repeating coils, etc., which will not be described in detail
here but will be referred to as the need arises.

Typical Station Layouts

Due to the various requirements for different types of service and
due in part to the different type of facilities, the general apparatus
layouts and arrangements at different repeater stations are not always
the same. Several of the more important general or typical layouts
will be briefly discussed, however.

On Fig. 15 is shown a layout of a typical intermediate station where
bridging is not required and where the gauge of the wires in the two
directions is the same. As may be noted from this figure, switching
facilities are provided so that the apparatus may be connected into
the circuit so as to properly take care of either the east-west or west-
east transmission. For this type of layout most of the apparatus is
common to both directions of transmission. The fixed artificial lines
or pads indicated by Note 1 on Fig. 15 are for the purpose of building
out whichever line has the lower 1,000-cycle attenuation so that this

368

BELL SYSTEM TECHNICAL JOURNAL

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM 369

line and associated pad will have the same 1,000-cycle loss as the
other line. As indicated, only one of these pads is required. This
building out of the shorter line minimizes attenuator adjustment when
the direction of transmission is reversed. The line amplifier in this,
as well as the other layouts to be discussed, is always set for a gain of
30 db.

On Fig. 16 is shown the layout of a typical intermediate non-
bridging station where the gauges of the wires on the two sides of the
repeater station are different. As mentioned earlier each gauge of
wire has its own particular low-frequency attenuation equalizer. Con-
sequently, where the gauges of the wires on the two sides of the
repeater station are not alike, it is necessary to arrange the station
layout so that the proper low-frequency equalizer will be associated
with the proper direction of transmission. This association of appa-
ratus may be readily observed from Fig. 16.

On Fig. 17 is shown the layout of a typical terminal station. This
layout differs from the intermediate station layout largely in the fact
that provision must be made for the introduction of predistortion
when the terminal station is transmitting a program to the open-wire
line and in the provision of a restoring network when the terminal
station is receiving a program from the open-wire line. The general
layout of the apparatus may readily be observed by reference to the
figure. The monitoring facilities at this type of station, in general,
differ from those provided at the normal intermediate station in that
a two-unit loudspeaker is provided for use as desired.

On Fig. 18 is shown the layout of a typical intermediate bridging
station where the gauge of the wires in the two directions is the same.
This arrangement differs largely from the arrangement shown on Fig.
15 in that the bridging amplifier is inserted immediately ahead of the
line amplifier so as to provide the necessary additional branches as
required. The general circuit arrangements involved to take care of
the different types of branches which may be encountered are indicated
on this figure. The photograph. Fig. 19, shows the program equip-
ment layout at an intermediate bridging station, which is of the type
just discussed in Fig. 18, utilizing, however, only one branch circuit
which is connected to a local broadcasting station.

In certain of the layouts just discussed, one apparatus unit desig-
nated as "Aux Filter" is shown which has not previously been men-
tioned. This is an 8,000-cycle low-pass filter somewhat similar to the
low-pass line filter, except that it is not designed to operate in parallel
with any high-pass filter. This filter is required at the transmitting
and receiving terminals, in the branches feeding the radio station and

370

BELL SYSTEM TECHNICAL JOURNAL

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

371

CO ll. (/) u. <

372

BELL SYSTEM TECHNICAL JOURNAL

'5o

12
'C

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

373

Fig. 19 — Intermediate bridging station bay layout.

374 BELL SYSTEM TECHNICAL JOURNAL

also in the high quality monitoring circuit to afford additional dis-
crimination against unwanted high-frequency interference as, for
example, interference from the carrier channels. This arrangement
of splitting the filter requirements enables a less expensive type of
line filter set to be employed.

Overall Performance

The initial application of this new program system was made on
two transcontinental circuits between Chicago and San Francisco.
One circuit, referred to as circuit 1, was routed through Omaha and
Denver over the central transcontinental line. The other circuit,
referred to as circuit 2, was routed via St. Louis and Kansas City to
Denver and thence over the same pole lead as circuit 1. The layout
of these two circuits is shown in Fig. 20. Circuit 1 was approxi-
mately 2,395 miles long and was routed through 17 repeater stations
involving 23 amplifiers in tandem. Circuit 2 was approximately
2,689 miles long and was routed through 19 repeater stations involving
29 amplifiers in tandem. Both circuits were routed through B-22
cable facilities between Sacramento and Oakland, California, and
non-loaded cable facilities in the transbay submarine cable between
Oakland and San Francisco.

At San Francisco a listening studio was set up in the Grant Avenue
office where the program circuits terminated. A two-unit loudspeaker
with suitable connecting networks was set in a 7' x T baffle, the
response of this loudspeaking system being practically uniform from
about 40 cycles to above 8,000 cycles. The room in which the loud-
speakers were located was acoustically treated so as to obtain the
proper reverberation time. A powerful amplifier having a flat gain-
frequency characteristic from 35 cycles to well above 8,000 cycles
supplied the loudspeaker system. A high quality phonograph system
for furnishing test programs was also installed at the Grant Avenue
office. The records used were of the vertical cut type and included
several recordings of a 75-piece orchestra as well as various solo and
instrumental recordings. Two outside pickup points were used, one
at the studios of one of the broadcasting companies at San Francisco
and the other at a hotel. At both of these places the moving coil
type of microphones was used and the latest type of high quality
pickup amplifiers. The pickup system used at both these places
had a response characteristic within about 2 db of being flat over the
range of 35 to 10,000 cycles.

Figure 21 is a photograph showing the special equipment placed in
the Grant Avenue offfce for carrying out the various overall tests and

ST. LOUIS

TERRE HAUTE

— HEV TO SYMBOLS —
[u] LINE AMPLIFIER FOR OPEN WIHE
[ba| bridging AMPUFIEfi FOR OPEN WIRE
0 LINE AMPLIFIER FOR CABLE
0 MONITORING AMPLIFIER

-ywC- VARIABLE ATTENUATOR

^WV FIXED RESISTANCE LINE

[l] LOW-FREQUENCT EQUALIZER

0 HIGH- FREQUENCY EQUALIZER

[nLc| NON-LOADED CABLE EQUALIZER

[eI LOADED CABLE EQUAU2ER AND
'-' ASSOCIATED PHASE CORRECTDB ETC

-ff" REPEATING COIL

' ' REPEATING COIL WITH LINE WINDINGS

J PARALLEL
[p] PREDISTORTING NETWORK
[r] RESTORING NETWORK
[i] 8000-CYCLE LOW-PASS LINE HLTER

0 AUXILIARY 8000-CYCLE
LOW-PASS FILTER

-<]] TWO-UNIT LOUD SPEAKER

— <] SINGLE-UNIT LOUDSPEAKER

f-nO SPECIAL HEADSET RECEIVERS AND
Cr*! ASSOCIATED NETWORK

@ VOLUME INDICATOR

NORMAL DIRECTION OF TRANSMISSION E-V

USES X CONNECTIONS; REVERSED W-£.

USES Y CONNECTIONS

Fig. 20 — Circuit layout for trial of wide-band open-wire program system.

WIDE-BAND OPEN -WIRE PROGRAM SYSTEM 375

Fig. 21 — Special apparatus bay layout.

WIDE-BAND OPEN -WIRE PROGRAM SYSTEM 375

Fig. 21 — Special apparatus bay layout.

376

BELL SYSTEM TECHNICAL JOURNAL

also shows the new equipment provided at San Francisco on the two
program circuits under discussion. The three right-hand bays accom-
modated the special equipment.

In making transmission measurements, the circuit under test was
first split up in a number of sections and each section was then meas-
ured at four test frequencies, namely, 50, 100, 1,000 and 7,000 cycles.
If the results were not within required limits the attenuators and
equalizers were readjusted as required. The various sections were
then connected together and the overall circuit measured at several
frequencies. Figure 22 shows the transmission-frequency character-

_X-

WERAGE OF 9 MEASUREMENTS
IXTREME DEVIATIONS

^^^.^

:=:^.

5s.

.^ ^

- -,

nrr^-'

^^^\;

■~-^S:^

^•^

1
1

'~ 30 50 100 500 1000 5000 10,000

FREQUENCY IN CYCLES PER SECOND

Fig. 22 — Transmission frequency characteristics of circuit No. 1, Chicago to San

Francisco.

istics of circuit 1. The solid line is the average of nine measurements
while the dashed lines show the extreme deviations obtained for any
of the nine measurements. Figure 23 shows corresponding data for

01 -6

/ERAGE OF 14 MEASUREMENTS
CTREME DEVIATIONS .

=^—E>

-__^'-

■

'•■

.--■"

t:::

2 2

t- 30 50 100 500 1000 5000 10,000

FREQUENCY IN CYCLES PER SECOND

Fig. 23 — Transmission frequency characteristics of circuit No. 2, San Francisco to

Chicago.

circuit 2. For comparison purposes the average characteristics of the
two circuits separately and the two of them connected in tandem
making a loop circuit of over 5,000 miles are shown on Fig. 24.

Other measurements were made to determine whether non-linear
effects were produced. For example, two frequencies were applied

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM

377

to the circuit, one being measured and the other alternately cut off
and on to determine whether one frequency adversely afTected the
transmission of the other or produced undesirable sum and difference
products. Such distortion effects were found to be small. Measure-
ments were made to determine whether the overall transmission varied
with the load applied. With a testing power which was varied in
magnitude from 50 milliwatts to .1 milliwatt, the transmission varied
slightly more than 1 db, that is, with the heavy load the circuit loss
was somewhat more than 1 db greater than at the light load.

A noise and crosstalk survey was made on these program circuits
and on message circuits on the same pole lead. Observations were
made at the terminals of the message circuits while a program was
being transmitted on the program circuits to determine the amount of

1 CIRCUIT NO. 2, SAN FRANCISCO TO CHICAGO, WEST

2 CIRCUIT NO. 1, CHICAGO TO SAN FRANCISCO, EAST

3 LOOP, SAN FRANCISCO TO CHICAGO TO SAN FRAN
2 WEST TO EAST, 1 EAST TO WEST

CIRCUIT NO. 2 2689 MILES 29 AMPLIFIERS

CIRCUIT NO. I 2395 MILES 23 AMPLIFIERS

LOOP 5084 MILES 52 AMPLIFIERS

MINUS VALUE INDICATES TRANSMISSION IS DOWN

FROM lOOO-CYCLE VALUE

TO EAST
TO WEST
CISCO,

:;:2^

100

500 1000

FREQUENCY IN CYCLES PER SECOND.

10,000

Fig. 24 — Average transmission frequency characteristics.

interference introduced into the message circuits from the program
circuits, and, conversely, observations were made on the program
circuits while various paralleling message circuits were in use, and the
resulting interference was recorded.

The noise or crosstalk volume on the program circuits was measured
by means of a volume indicator, which had inserted between it and the
circuit at the point of measurement a network having a loss-frequency
characteristic such that the various frequencies affecting the meter
reading were attenuated or weighted in much the same way that the
ear weights the different frequencies. Crosstalk volume and noise on
the message circuit were measured with an indicating meter in much
the same manner except that the network used here had an attenuation
frequency characteristic corresponding very nearly to that of the ear
and an average telephone set. The network used on the program

378 BELL SYSTEM TECHNICAL JOURNAL

circuits was referred to as a "program weighting network," while
that used with the message circuit was the ordinary "message weight-
ing network." The noise and crosstalk volume was then recorded in
db referring to reference noise with either program weighting or
message weighting. Reference noise is that amount of interference
which will produce the same meter reading as 10~^^ watt of 1,000-cycle
power, which is 90 db below 1 milliwatt.

The results of this survey indicated that in consideration of the
layout and levels of the existing message circuits and of the noise
existent on these circuits and on the program circuits, the value for
maximum program volume, should, under normal conditions, be + 3
referred to reference volume; that is, at this value the best balance
between program to message crosstalk and program circuit noise
would result. It was also determined that on very long sections, or
on sections where all circuits were subjected to severe noise exposure,
the maximum volume on the program circuits could be increased 3 db
to improve the signal-to-noise ratio on the program circuits. This
higher volume could be permitted in these cases since on the longer
sections the message circuits also usually operate at higher levels,
and on the especially noisy short sections the increased crosstalk to
the message circuits will ordinarily be masked by the greater noise.

The average noise measured at San Francisco or Chicago at the
circuit terminals at the reference volume point was 49 db above
reference noise "program weighting" when the restoring network
was included at the receiving terminal. The noise averaged 5 db
higher than this with the restoring network removed. This value of
noise is about 43 db below the maximum power of the program
measured at the same point with the same measuring instrument.
This, therefore, establishes a signal-to-noise ratio of about 43 db,
thus permitting a volume range of approximately 40 db.

The various tests referred to gave statistical data concerning the
transmission performance of the circuits from which it could readily
be predicted that the circuits would transmit programs with very little
impairment to quality. To substantiate this, very critical listening
tests were made, comparing the quality of a program after it had been
transmitted over various length circuits with the same program trans-
mitted over a reference circuit which was distortionless over the
frequency range for which the circuits were designed, namely, to 8,000
cycles. Figure 25 shows schematically the terminal arrangements
employed at San Francisco for these listening, or, as they are more
commonly called, comparison tests.

Various types of programs were used, such as speech, vocal and

WIDE-BAND OF EN-WIRE PROGRAM SYSTEM

379

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380 BELL SYSTEM TECHNICAL JOURNAL

instrumental selections, orchestral renditions, both classical and jazz.
Quite a number of observers were employed, some of whom were
present on several tests and a few on all tests. On tests made on a
San Franciscb-Denver-San Francisco loop involving 2,600 miles of
circuit, no observer was able to consistently differentiate between the
quality over the reference circuit and that over the program circuit.
On tests made on the San Francisco-Chicago-San Francisco loop
certain of the more experienced observers were able to differentiate
between the circuits somewhat more than 50 per cent of the time, but
this, it must be remembered, was on a direct comparison test. None
of the observers could tell with any assurance which was the program
circuit and which was the reference circuit if a few minutes were
allowed to elapse between switches. On the Chicago loop 264 obser-
vations were made on direct comparison tests on which 60 per cent of
the observations favored the reference circuit and 40 per cent favored
the program circuit.

Included as part of the overall program, were tests to determine the
volume range, maximum volume obtainable and speed with which
the circuits could be reversed.

On the volume range tests a source of program was obtained and
so regulated that it had a very narrow volume range. This was then
applied to the circuit with the sending end gain adjusted so that the
maximum volume applied at the repeater outputs was + 6. The
sending end gain was then gradually decreased so as to apply a
gradually decreasing volume to the circuit. This process was con-
tinued until the program volume was so weak that the line noise
interfered with its satisfactory reception. The amount that the send-
ing end gain was adjusted determined the volume range. The average
value for several tests was slightly in excess of 40 db. The maximum
volume was determined by switching a 10 db pad from the sending
end to the receiving end of the circuit and listening to a transmitted
program, noting the point at which there was a quality difference
between the high volume and low volume condition. It was found
that a slight difference could be detected when the maximum volume
on the high volume condition was -\- 10, thus showing the circuit was
capable of handling a maximum volume slightly lower than this value.

As mentioned earlier, switching means are provided at each station
for reversing the direction of transmission. On the initial field tests
it was demonstrated that the circuits could be reversed readily and
at the same time maintain satisfactory overall characteristics. At the
present time, on receipt of proper advance notice, the circuits are
being reversed on commercial programs in approximately 30 seconds.

WIDE-BAND OPEN-WIRE PROGRAM SYSTEM 381

Conclusion

The above development provides a program transmission system
applicable to open-wire lines which even for very long distances will
provide transmission characteristics which should be adequate for
program transmission for a number of years to come.

Acknowledgment

The author makes grateful acknowledgment to his associates in
the Bell Telephone Laboratories, to members of the Long Lines
Department of the American Telephone and Telegraph Company,
and of the Pacific Telephone and Telegraph Company, for their
cooperation in connection with the setting up of the circuits and
participation in many of the tests, and to the National Broadcasting
Company and the Columbia Broadcasting System for their assistance
in making available the program pickup sources at San Francisco.

Line Filter for Program System *

By A. W. CLEMENT

Open wire circuits recently have been developed for transmitting radio
broadcast programs with greater naturalness and over greater distances
than heretofore.^ The simultaneous utilization of these circuits for the
transmission of broadcast programs and carrier telephone messages requires
the use of line filters to restrict the program and carrier currents to the
proper circuits. The low pass line filter developed for the program circuits
and its contribution to the maintenance of good quality in the programs
transmitted are described in this paper.

PROGRAM transmission systems operated on open wire telephone
lines ordinarily are not assigned the exclusive use of the lines,
but usually share them with other communication facilities. The
wide-band system described in an accompanying paper ^ transmits
currents in the frequency band extending from 35 to 8,000 cycles per
second, while the lines over which it is routed possess useful transmis-
sion ranges extending from 35 to considerably above 30,000 cycles.
In order that the range above 8,000 cycles shall not be wasted, carrier
telephone systems utilizing these frequencies usually are operated on
the same wires with the program systems.

Line filters are used at each terminal and repeater point in the pro-
gram system to separate the program currents from the carrier currents
and to guide each to the proper channel. They are operated in sets
consisting of a low-pass filter and a high-pass filter connected in parallel
at one end, the end that faces the line. The low-pass filter transmits
the program currents freely while effectively excluding the carrier
currents, and the high-pass filter transmits the carrier currents while
excluding the program currents.^

The line filters are located in the open-wire program systems as
shown in Figs. 1, 15, 16, 17, 18, and 20 of the accompanying paper by
H. S. Hamilton.^ The low-pass filter is in the direct path of the
program currents and therefore has a number of features of special
interest. It is the object of this paper to describe this filter and its
contribution to the maintenance of good quality in the programs trans-
mitted over the system.

* Published in April, 1934 issue of Electrical Engineering, Scheduled for presen-
tation at Pacific Coast Convention of A. I. E. E., Salt Lake City, Utah, September,
1934.

^ "Wide-Band Open- Wire Program System" by H. S. Hamilton, published in this
issue of the Bell. Sys. Tech. Jour.

^ "Telephone Transmission Networks" by T. E. Shea and C. E. Lane, published
in ^. /. E. E. Transactions, Vol. 48, 1929, pages 1031-1034.

382

LINE FILTER FOR PROGRAM SYSTEM 383

This low-pass line filter, with its associated high-pass filter, makes it
possible to use the open-wire lines simultaneously for wide-band
program service and for commercial carrier telephone service, without
impairing the quality of the program. It represents an improvement
over older types of line filters, as well as an advance in the technique
of equalization in filters. In cases requiring careful delay and loss
equalization, it has been the usual practice to design the filter first
to supply the required discrimination or filtering action, and then
design a delay corrector to correct for the delay distortion in the
filter, after which a loss equalizer is designed to correct for the ampli-
tude distortion in both the filter and the delay corrector. The loss
equalizer introduces a small delay distortion which usually can be
anticipated and corrected in the delay corrector. In the wide-band
program filter the functions usually performed by these three separate
types of networks have been combined, with a consequent saving in
cost and space.

Requirements to Be Met by Program Filter

To function effectively as a line filter, the low-pass filter must
provide sufficient discrimination against carrier currents to make their
effect completely inaudible in all the receivers connected to the pro-
gram system. Discrimination varying from 46 to about 90 db is
necessary to accomplish this end. Because of the presence of an
auxiliary low-pass filter ^ which supplies considerable loss in the fre-
quency ranges where the requirement is unusually severe, each line
filter need furnish discrimination varying only from 40 to about 60 db.

From the standpoint of program quality, it is essential that the
line filter, while furnishing the foregoing discrimination, shall not
introduce any appreciable distortion into the program. This require-
ment would call for nothing unusual in the way of filter design if there
were only a few filters in the system. Long open-wire program
systems, however, may extend as far as 3,000 or 4,000 miles, and may
contain as many as 50 low-pass line filters. A program that has
traversed such a circuit still must be comparable in quality to a
program that is broadcast from the point at which it originated.
Since the system contains much other apparatus, such as equalizers
and amplifiers, each low-pass line filter can be permitted to introduce
not more than about 1/100 of the distortion that can be tolerated
in the whole system, assuming 50 filters in the system.

There are two types of distortion that must be controlled very
carefully in the program filter: these are (1) amplitude distortion, and
(2) delay, or phase, distortion. Amplitude distortion is introduced

384 BELL SYSTEM TECHNICAL JOURNAL

by a filter when its loss is not the same at all frequencies in the trans-
mitted band, currents of some frequencies being attenuated more than
others. The effect of amplitude distortion on the program is to change
the relative intensities, or volumes, of tones of the frequencies at which
distortion occurs, thus impairing the naturalness of the program.
Amplitude distortion ordinarily can be corrected without much diffi-
culty by means of suitable attenuation equalizers.

Delay distortion is introduced by a filter when different frequency
components of a signal require different lengths of time for propagation
through the filter. This type of distortion is related directly to the
shape of the phase shift-frequency characteristic. The slope of this
phase shift curve usually is taken as a measure of the delay introduced
by the filter. Stated mathematically, the delay in seconds is taken as
dB/do), where B is the phase shift in radians and w is 2-n-f, f being the
frequency in cycles per second. Thus if the phase shift of the filter
is proportional to frequency, dBJdoo, or the delay, is constant and
there is no delay distortion. In this case the wave form of a signal
transmitted through the filter remains unchanged, the signal being
delayed in transmission an interval of time corresponding to the slope
of the phase shift curve. If the slope of this curve is not constant
over the transmitting band of the filter, however, delay distortion is
introduced. In low-pass filters, the difference between the slope of
the phase shift curve at a given frequency and the minimum slope of
the curve is a measure of the delay distortion at that frequency.

A discussion of delay distortion in telephone apparatus, including
filters, as well as a discussion of the effect of delay distortion on tele-
phone quality, may be found in two recent articles on these subjects.^' ^
Whereas the effect of amplitude distortion is to weaken or strengthen
some of the tones in the sound being transmitted with respect to the
other component tones, the effect of delay distortion is to introduce
unnatural audible effects which may become so pronounced as to be
annoying if the delay distortion be great enough.

Delay distortion is present in most filters used in communication
work, but ordinarily not in such magnitude that its effect is noticeable.
As a rule, it need be considered only when a large number of filters is
used in a single circuit, as in the case of the program systems. Delay
distortion is in general more difficult to correct than amplitude distor-
tion. One of the unusual features of the low-pass line filter used in
the wide-band program circuits is the means employed to keep it
free from delay distortion.

^" Phase Distortion in Telephone Appiiratus" by C. E. Lane, Bell. Sys. Tech.
Jour., July, 1930.

* "Effects of Phase Distortion on Telephone Quality," by J. C. Steinberg, Bell
Sys. Tech. Jour., July, 1930.

LINE FILTER FOR PROGRAM SYSTEM

385

The filter consists of four parts, each with distinguishing functional
characteristics. The separate parts, or sections, have image im-
pedances such that when they are joined together no current is
reflected at the junctions. Figure 1 shows the filter in block sche-

IMPEDANCE
CORRECTING
TERMINATION

PEAK
SECTION

DELAY

AND LOSS

EQUALIZING

LATTICE

SECTION

IMPEDANCE
CORRECTING
TERMINATION

Fig. 1 — Block schematic diagram of filter.

matic form. Each part of the filter provides some of the attenuation
required to exclude carrier currents from the program circuit, the
attenuation of the complete filter being the sum of the attenuations
of all parts. On Fig. 2 are shown the loss-frequency characteristics
of the various sections and of the complete filter.

O 50

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.^''''

"''

■^5-^''''

\t-

C'^^^'"'

£ SECTION

0 2 4 6 8 10 12 14 16 18 20 22 24

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 2 — Loss in filter and in component sections.

Delay Equalization

Likewise, the phase shift of the complete filter is the algebraic sum
of the phase shifts of all sections. The phase shift of the filter exclu-
sive of the delay and loss equalizing section is similar to that of the

386 BELL SYSTEM TECHNICAL JOURNAL

usual ladder type low-pass filter. Over the lower frequencies of the
transmitting band the phase shift-frequency characteristic is prac-
tically linear with frequency, but at the higher frequencies the slope
of this curve increases gradually with frequency and becomes very
large near the upper edge of the band. Phase shift varying in this
manner introduces much more delay distortion than can be tolerated,
and therefore has to be corrected. It is one of the functions of the
delay and loss equalizing section, which is of the lattice type, to correct
for this distortion. The phase shift of this lattice section is such that
when it is added to that of the rest of the filter the total phase shift is
very nearly proportional to frequency over the whole program band,
and delay distortion thus is almost entirely eliminated.

The property of the lattice section by which its phase shift can be
made to vary with frequency in the desired manner is expressed in the
following characteristic equation, which holds only in the transmitting
band and when the section is terminated in its image impedances:^

Kf 1

In this equation, B is the phase shift in radians;/ is the frequency in
cycles per second ; /i, /2, /s, and fc are frequencies at which the phase
shift of the section is successive multiples of tt radians or 180 deg.,
fc being also the cut-off frequency of the filter; and i^ is a constant
controllable by assigning the proper values to the coils and condensers
of the section. By assigning to/i, /2, and/3 the values of frequency at
which it is desired that the phase shift of the section shall be tt. It,
and Stt radians, respectively, and by giving K the proper value, the
phase shift-frequency curve is made to approximate the ideal one
which completely would correct the delay distortion of the filter.
Figure 3 illustrates the building up of the phase shift characteristic.
The delay corresponding to the rate of change of the phase shift with
frequency is plotted in Fig. 4. The average delay introduced by the
filter is about 0.00035 sec. It may be noted that for frequencies
below 7,500 cycles per second, the variation from this average does
not exceed 0.000025 sec. Thus the delay due to 50 filters in a long
program circuit does not deviate from the average in this frequency
range by more than 0.00125 sec. Distortion of this amount ordinarily
would not be detected by the average listener. Above 7,500 cycles

5 U. S. Patent No. 1,828,454 to H. W. Bode.

LINE FILTER FOR PROGRAM SYSTEM

387

O 600

I 500

200

COMPLETE
FILTER

^ LATTICE
SECTION

FILTER

LESS LATTIC

SECTION

:e y

0 123456789

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 3 — Phase shift in fiher and in component parts.

per second the delay gradually increases with frequency, rising quite
rapidly outside the program band. The high attenuation at fre-
quencies above the program range, however, eliminates any effect this
distortion otherwise might have on the program.

513

O 500

Q 5

—

— -

—

2 3 4 5 6

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 4 — ^Delay-frequency characteristic of filter. The ordinates of this curve are
proportional to the slope of the upper curve of Fig. 3.

388 BELL SYSTEM TECHNICAL JOURNAL

Loss Equalization

Another function of the lattice section is to make the loss of the
filter constant in the program frequency band. In a dissipationless
filter terminated in its image impedances (which is substantially the
condition under which this filter is operated) the loss in the trans-
mitting band is zero. The effect of dissipation is to introduce a loss
which is given approximately in this band by the equation:

where Ad is the loss due to dissipation, B is the phase shift of the non-
dissipative filter, and Q is the average dissipation factor of the coils
(dissipation in the condensers being negligible, ordinarily). The
factor Q is equal to the average of the ratios o^Le/Re, and o}/2Q in
equation (2) therefore may be written Re/2Le, where Re and Le are
the effective resistance and effective inductance, respectively, of the
coils.

In the coils of the program filter, Q is about proportional to fre-
quency over the lower portion of the program band, but above this
range the factor co/lQ increases with frequency. For the filter exclu-
sive of the lattice section, the factor dB/dco is also greatest at the higher
frequencies, as may be seen from the lower curve in Fig. 3; hence this
part of the filter introduces much more amplitude distortion than is
permissible. For the lattice section alone, however, the factor dBjdw
is greatest at the lower frequencies, as is apparent from the middle
curve of Fig. 3. Thus the natural tendency of dissipation in the
lattice section is to compensate for the distortion in the other sections
of the filter. This compensating tendency can be controlled to a
considerable degree, since by equation (2) ^d is proportional to Re.
By proper adjustment of the effective resistance of the coils of the
lattice section, its loss is made practically complementary to that of
the rest of the filter, so that the loss of the complete filter is sub-
stantially constant throughout the program range.

The loss of the filter in the transmitting frequency band is shown in
Fig. 5. The average loss below 7,000 cycles per second is about 0.53
db and the deviation from this average does not exceed 0.03 db.
Considering again a circuit containing 50 filters, the deviation from
the average loss introduced by the filters does not exceed 1.5 db in
this range. Between 7,000 and 7,500 cycles per second the amplitude
distortion per filter is about 0.10 db, and above 7,500 cycles the loss
increases in such a way as to tend to mask the small delay distortion
in this range.

LINE FILTER FOR PROGRAM SYSTEM

389

o<^

or,

-O0.8

0.4

0 I 2 3 4 5 6-7 8

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 5 — Loss of filter in program frequency band.

Impedance Correction

In the discussion of the lattice section it was stated that its phase
shift is given by equation (1) only when the section is terminated in
its image impedance. To facilitate the design and simplify the filter
structure, this section has been given an image impedance of the
simplest type. This impedance, Z/, varies with frequency according
to the following equation :

^^ = -4=' (3)

1 -^-

where Zo is the "nominal impedance" of the filter, a constant equal
approximately to the average impedance of the open-wire lines in the
program band; and fc is the theoretical cut-off frequency. Thus the
image impedance rises with increasing frequency to a very high value
near the cut-ofif; and, since the line impedance is practically constant
except at very low frequencies, a large mismatch would result at the
upper edge of the transmitted band if the lattice section were con-
nected directly to the line. The impedance correcting sections at
the ends of the filter are employed to avoid this mismatch. The
properties of these sections are such that when they are inserted
between the lattice section and the line or the office terminating
apparatus, the impedance of the filter matches that of the line and
the office apparatus, and the lattice section faces its own image im-
pedance. In this manner, both internal and external reflections largely
are avoided ; and the phase shift of the lattice section has the proper
value. ^

The general theory on which the design of the impedance correcting
sections is based is discussed at length in a recently published article.''
In brief, the sections consist of two parts: a 4-terminal network to

^ "Impedance Correction of Wave Filters," by E. B. Payne, Bell. Sys. Tech. Jour.,
October, 1930.

^"A Method of Impedance Correction," by H. W. Bode, Bell. Sys. Tech. Jour.,
October, 1930,

390

BELL SYSTEM TECHNICAL JOURNAL

make the resistance of the filter approximately constant over the
program band, and a 2-terminal network placed in shunt at the end
to cancel the reactance of the filter in this band. The inductance and
capacitance of the coils and condensers of the 4-terminal network are
related to the coefficients of a power series expansion of the right-hand
part of equation (3) in the manner explained in the article by H. W.
Bode.'' The 2-terminal shunt network at the apparatus end is de-
signed so that, while canceling the reactance of the filter in the program
band, it resonates just above the band to produce a peak or sharp
maximum of attenuation. It thus supplies the sharp selectivity
required to produce an abrupt change from free transmission of the
program frequencies to high attenuation of the carrier frequencies.

°-^«^nti^fcii

TO LINE
AND TO :
HIGH-PASS
FILTER

IMPEDANCE •— IH
CORRECTING „''
SECTION i^tAt\

DELAY AND LOSS EQUALIZING
LATTICE SECTION

Fig. 6 — Schematic diagram of filter.

At the line end, the impedance correcting section is designed for
parallel connection with the high-pass line filter. The high-pass filter
itself acts as the shunt reactance-canceling network.

The peak section shown at the left of the delay and loss equalizing
section in Fig. 1 provides attenuation which rises rapidly with fre-
quency above the program band in such a way as to add to the selec-
tivity of the filter. It is a ladder section of a type often employed in
filters for its selectivity.

The filter is designed to match the average impedance of the open-
wire lines. The impedance of the office apparatus, however, is slightly
higher than that of the lines and the filter. An autotransformer
therefore is used at the end of the filter connected to the office appa-
ratus, to effect the required change in impedance. A schematic
diagram of the complete filter is shown on Fig. 6, the parts being
marked for identification in accordance with the foregoing discussion.

Contemporary Advances in Physics, XXVIII
The Nucleus, Third Part *

By KARL K. DARROW

Transmutation, the major subject of the Second Part of this sequence on
the nucleus, assumes again a leading role in the present article. Remark-
able cases have been discovered since the first of the year, including a great
number in which the impact of one nucleus upon another (or of a neutron
"on a nucleus) provokes an instantaneous transmutation which is followed
after seconds, minutes or hours by the spontaneous breaking-apart of one
of the resultant nuclei. One may say that these last are the nuclei of new
kinds of radioactive elements, and the phenomena are often called " induced
radioactivity"; but many of these new unstable elements differ from all
radioactive .bodies hitherto known in that they emit positive electrons.
Some additional examples of transmutation are described at the end of
this article.

Induced Radioactivity

UP to the end of last year (1933) it was taken for granted that
transmutation is practically instantaneous: that when two nuclei
collide, the ensuing fusion and disruption (if any there be) are ended
within a time inappreciably short. Nowadays, however, many cases
are being discovered, in which a disruption occurs a long time —
several minutes or even hours, possibly not for days — after the collision.
We must suppose that at the moment of the collision something
happens, which entails the eventual disruption. In a very few cases
we may be reasonably sure that this initial "something" is itself a
transmutation, resembling those previously known in that it is
instantaneous, but differing from them in that one of the resulting
fragments is an unstable nucleus, of which the eventual spontaneous
disruption is that which is observed. This may be the course of
events in all cases, but it is also conceivable that in the collision one
of the original nuclei may be put into an unstable state without the
occurrence of an initial transmutation.

The first-to-be-known of these phenomena was discovered by M.
and Mme. Joliot at the very start of 1934, when they exposed samples
of aluminium (and boron and magnesium) to the bombardment of the
5.3-MEV alpha-particles from polonium, and after a few minutes of
exposure removed them from the bombarding beam and placed them

* In this issue is published the first section of "The Nucleus, Third Part." The
paper will be concluded in the October, 1934 issue.

"The Nucleus, F"irst Part" was published in the July, 1933 issue of the Bell Sys.
Tech. Jour. (12, pp. 288-330), and "The Nucleus, Second Part" in the January,
1934 issue (13, pp. 102-158).

391

392

BELL SYSTEM TECHNICAL JOURNAL

beside a Geiger-Miiller counter.^ Hundreds of counts per minute
disclosed the emergence of fast-flying particles from the samples.
The number per minute fell off exponentially (Fig. 1) with the lapse of
time: a very important feature, for this is the law of radioactivity.
The exponential decline implies that the nuclei which were destined
to emit these particles were formed at the moments of collisions and
existed intact for periods of time — "lifetimes" — not the same for all
but distributed in a perfectly random fashion. Such a decline is

2.0
1.9
1.8

1.7

<n 1.6

t-
Z 1.5

i 1

— ■

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0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16

TIME IN MINUTES

Fig. 1 — Exponential decay of the radioactivity induced by boron, aluminium,
magnesium with alpha-particles: semi-logarithmic plot. (F. Joliot & I. Curie-
Joliot, Journal de Physique.)

characterized by a singe constant, the "half-period," or lapse of time
during which the rate of emission of particles drops to one-half of its
initial value. The half-periods in the three cases examined by the
Joliots are different: boron 14', magnesium 2' 30", aluminium 3' 15".
This is a welcome feature wherever it occurs, as when two substances
exhibit different half-periods the effect cannot be ascribed to any
contamination common to both.^

Since thus there are not only delays between the bombardment and
the ultimate disruption, but also (at any rate in the tested cases) a

1 "The Nucleus, Second Part," p. 119; the "Geiger-Miiller" counter has a thin
wire for its inner electrode, while most of those called simply " Geiger counters" have
needle-points, though the earliest counters invented by Geiger were of the former

2 Cases are on record in which several different elements have exhibited decay-
curves, each the sum of two exponentials, one having a half-period characteristic
of the element and the other a half-period common to all samples; the latter is then
ascribed to a common admixture.

CONTEMPORARY ADVANCES IN PHYSICS 393

random distribution of the lengths of these delays, it is customary
and proper to refer to these phenomena as "induced radioactivity."

Examples of induced radioactivity have already been provoked with
all of the four known agents of transmutation: alpha-particles acting
on B, Na, Mg, Al and P, — protons acting on boron and carbon —
deutons acting on boron and carbon and a number of othens — neutrons
acting on a large variety of elements. The half-periods reported when
neutrons are the agents have ranged from a few seconds to a couple
of days, while in all other cases they are of the order of a few minutes.

The nature of the ejected particles resulting from the ultimate
disruption is of course of the greatest importance. The Joliots found
them to be positive electrons or orestons ^ in their pioneering experi-
ments, and this was confirmed by Ellis and Henderson at the Cavendish
Laboratory; the tests have been made by applying magnetic fields to
tracks made visible in the Wilson chamber or to beams of particles on
their way to photographic or other detectors, and are doubtless to be
regarded as conclusive, though no details have yet been published.
Induced radioactivity provoked by a-par tides, in the few cases so far
known, thus results in the emission of orestons.^ This seems also to
be the rule when it is provoked by deutons or protons, as is shown by
splendid Wilson-chamber photographs (Figs. 2, 4) obtained by
Anderson when samples of various elements (boron in the form of
B2O3, carbon, aluminium, beryllium) were first bombarded for several
minutes and then put right into the chamber itself. The tracks of the
particles springing from the samples have the specific aspect of electron-
tracks,^ and in the imposed magnetic field of 800 gauss they have a
curvature of which the sense proves the particles to be positive. On
the other hand it is stated by Fermi that the radioactivity induced
by impacts of neutrons involves the emission of negative electrons,
though in his very brief reports there is no intimation as to how this
is shown.

For each individual case it is important to inquire whether the
half-period is independent of such circumstances as the kinetic energy
Ka of the impinging particles. If so, it is sufficient to postulate a
single kind of unstable nucleus resulting from the collisions; otherwise,
not. This has been investigated in the cases of radioactivity induced
by alpha-particle impact; the Joliots reduced Kq from 5.3 to 1 MEV,
without observing any change in the half-perod.

^ As an occasional alternative to "positive electron" I adopt Dingle's beautiful
word "oreston" (Orestes, in Greek mythology, was the brother of Electra).

* Excepting that the Joliots have lately reported that magnesium emits electrons
of both signs, which they attribute to different isotopes.

^ "The Nucleus, First Part," p. 303.

394

BELL SYSTEM TECHNICAL JOURNAL

■-^ \\

; ^.:

Fig. 2 — Induced radioactivity.resulting from bombardment of carbon by 0.9-MEV
protons: tracks of positive electrons. (C. D. Anderson.)

25 r

0.8 1.0

MEV

Fig. 3 — Distribution-in-energy of positive electrons of the induced radioactivity
resulting from bombardment of carbon by 0.9-MEV protons. (Anderson & Nedder-
meyer, Physical Review.)

CONTEMPORARY ADVANCES IN PHYSICS

395

One next inquires whether all of the orestons resulting from a given
type of impact spring off with the same energy. Experience with
natural radioactivity shows that while alpha -particles are emitted
either with a single definite energy or with one of several definite
discrete energies characteristic of the particular process, negative
electrons (beta-particles) are always emitted with a very wide and

F"ig. 4 — Induced radioactivity resulting from bombardment of boron oxide by
0.9-MEV deutons: tracks of positive electrons, some springing from gas adjoining
the target, as though a radioactive gas had diffused out of the boron oxide block.
(Anderson.)

continuous distribution-in-energy. Short as is the time which has
elapsed since January last, and weak as are the beams of positive
electrons resulting from induced radioactivity, it is already assured
that in several cases at least it is the latter rule which is followed and
not the former. The best distribution-curves are those derived at
Pasadena from a statistical study of oreston-tracks made visible in a
Wilson chamber and curved by an imposed magnetic field; they refer
to radioactivity provoked by 0.9-MEV protons falling on carbon, and
by 0.9-MEV deutons falling on Be, B, C and Al. I reproduce one of
these curves as Fig. 3 (another curve obtained with 0.7-MEV protons

396 BELL SYSTEM TECHNICAL JOURNAL

falling on carbon is indistinguishable from it). In one of the cases of
radioactivity induced by alpha-particle impact, Ellis and Henderson
at the Cavendish Laboratory observed a continuous distribution of
energies of the positive electrons ranging between 1 and 2.5 MEV.

In all of these cases of delayed transmutation, nothing is observed
of the ultimate disruption excepting the emergence of the electron;
the other fragments apparently do not receive energy enough to make
a track or reach a detector, and our knowledge is thus forcedly incom-
plete as it is with most other examples of transmutation. In respect
to the initial process occurring at the collision, the prospect of attaining
complete knowledge seems even dimmer. We are not without some
guidance, for when alpha-particles impinge on aluminium or boron,
certain particles are expelled with apparently no delay, and these may
be fragments resulting from that initial process. There is, however,
an embarras de choix; both protons and neutrons are expelled in each
of these cases; if one is a fragment resulting from the same process of
which an unstable nucleus of half-period 3' 15" is another fragment,
then the other must be due to something entirely different. Actually
Ellis and Henderson inferred from their data that in the case of
aluminium, the number of protons produced by a given bombardment
is fifty times as great as the number of unstable nuclei which eventually
eject orestons. This obliges us to assume that the initial process out
of which the delayed transmutation arises is either the one which
produces the neutrons, or else some other producing no fast-moving
particle at all.

Decision between these alternatives is made from a most notable
experiment of the Joliots, sufficient indeed by itself to settle the nature
of the initial process. To introduce it in the way in which it suggested
itself to them, I make the tentative assumption that the initial process
is a case of what is called ^ "disintegration by capture with emission
of a neutron," and that the residue of this process is the unstable
nucleus. Embodying this assumption in equations of "nuclear
chemistry" written after the fashion of those in the Second Part with
atomic number for a subscript preceding the symbol of each element
(so that ow and le become the proper symbols for a neutron and an
oreston) we have for boron and for aluminium:

2He + aB = 7N + ow, followed by 7N = eC + le,
2He + 13AI = 15P + 0", followed by 15P = uS\ + le.

The unstable nucleus, if it is surrounded by its proper quota of orbital
6 "The Nucleus, Second Part," pp. 147-148, 155.

CONTEMPORARY ADVANCES IN PHYSICS 397

electrons, should then possess the chemical properties of nitrogen in
the former case, phosphorus in the latter.

The important experiment of the Joliots consisted in showing that
when a sample of boron (or aluminium) is first exposed to alpha-
particle bombardment and then to such chemical processes as would
remove nitrogen (or phosphorus) commingled with the boron (or
aluminium), the induced radioactivity is itself removed and carried
away. I quote verbatim: "We have irradiated the compound BN.
By heating boron nitride with caustic soda, gaseous ammonia is
produced. The activity separates from the boron and is carried away
with the ammonia. This agrees very well with the hypothesis that
the radioactive nucleus is in this case an isotope of nitrogen. When
irradiated Al is dissolved in HCl, the activity is carried away with the
hydrogen in the gaseous state, and can be collected in a tube. The
chemical reaction must be the formation of PH3 or SiH4. The
precipitation of the activity with zirconium phosphate in acid solution
seems to indicate that the radio-element is an isotope of phosphorus."

The assumed equations are thus substantiated in a very striking
way. These experiments are in a sense the first chemical identifica-
tions of any product of transmutation; I say "in a sense," because
while this nitrogen and this phosphorus are identified by virtue of
chemical properties, they are detected only by virtue of their radio-
activity.^

Some striking photographs, taken at Pasadena with an expansion-
chamber containing a block of boron oxide previously bombarded by
alpha-particles, show many tracks of positive electrons springing from
points in the air of the chamber (Fig. 4). It is inferred that the
unstable nuclei formed from the boron (not from the oxygen, since
bombardment of Si02 has no effect) are carbon nuclei which unite
with electrons to form carbon atoms and then with oxygen atoms to
form molecules of CO or CO2 having a natural tendency to diffuse out
of the solid mass. The radioactivity may be driven completely out
of the solid block in short order by heating to 200° C. The radioactive
particles are unable to pass through a liquid-air trap.

" Inserting mass-numbers into the equations, one finds that since Al has but the
one known isotope 27, the value 30 is indicated for the mass-number of "radio-
jihosphorus," as Joliot calls it; while since boron has two isotojies 10 and 11, the two
values 13 and 14 are indicated for radio-nitrogen, with no certain evidence to dictate
a choice between them. Ordinary stable phosphorus has no known isoto]^e 30, and
ordinary stable nitrogen has no known isotope 13, but the vast majority of its atoms
are of mass-number 14. It seems natural that a very unstable isotope should have
a different mass-number from any of the known and stable ones, and this may be a
valid argument for inferring that it is B'" rather than B'' wh'ich is concerned in the
induced radioactivity of boron; but there is nothing to prohiliit us from su])posing
that there may be an unstable isotope of nitrogen agreeing in mass-number with
the one which is durable.

398 BELL SYSTFM TECHNICAL JOURNAL

The result of bombarding carbon with deutons might be expected
to be the same as that of bombarding boron with alpha-particles, it
being natural to assume the reactions:

,W + 6C12 = ^Ni3 ^ ^^1^ followed by 7N" = ,0' + ic

The half-period of the delayed disruption has been determined at
Pasadena as 10.3 minutes. This does not agree with that observed
by the Joliots when alpha-particles are projected against boron. The
disagreement is not so welcome as agreement would have been, but
does not in the least invalidate the foregoing equations, since it is
perfectly conceivable that two different unstable nuclei with different
half-periods might both have the atomic number 7 and the mass-
number 13. Bombardment of carbon with protons leads to delayed
disruptions with the same half-period of about ten minutes, and this
is not so easy to understand as it may seem, since the obvious notion
that the proton and the C^^ nucleus simply merge into a nucleus N^^
which later on explodes leads into difficulties with the principles of
conservation of energy and conservation of momentum.

As to the way in which the number of observed disruptions varies
with the kinetic energy Ko of the impinging particles, there are data
relating to the bombardment of aluminium by alpha-particles. The
Joliots varied Kq from 5.3 MEV downwards; they report that the
number of positive electrons diminishes with falling Ko, becoming
imperceptible for boron at about 3 MEV, for Mg and Al at 4 to 4.5
MEV. Ellis and Henderson varied Kq from 5.5 upward to 8.3 MEV,
by using alpha-particles emitted from other radioactive bodies than
polonium; they found the number of orestons steadily increasing with
rising Kq, rising in the ratio 15 : 1 as Xo was raised from 5.5 to 7 MEV,
and showing signs of approaching a maximum not far beyond Ko
= 8.3 MEV.

The positive electrons emitted in induced radioactivity are fre-
quently— perhaps generally — accompanied by high-frequency photons,
of which energy-measurements may hereafter show that they are due
to the coalescence of positive with negative electrons to form light.

I close this section by listing the elements which have been observed
to display induced radioactivity after bombardment by one or other
of the four agents of transmutation, and add those which have been
tested without positive results, in order to show the scope of the
experiments. In certain cases positive results have been obtained by
some observers and not by others, but this may signify simply a weaker
bombarding stream or a less sensitive detector in the apparatus of
the latter.

CONTEMPORARY ADVANCES IN PHYSICS 399

Bombardment by alpha- par tides : B, Mg, Al (JoHots, Ellis & Hender-
son); Na, P (Frisch) ; negative results with H, Li, Be, C, N, O, F,
Na, Ca, Ni, Ag (Joliots).

Bombardment by deutons: Li, Be, B, N, C, O, F, Na, Mg, Al, Si,
P, CI, Ca (Henderson, Livingston & Lawrence, with 3-MEV deutons);
Li, Be, B, C, Mg, Al (Crane & Lauritsen, with 0.9-MEV deutons).

Bombardment by protons: B, C (Crane & Lauritsen); C (Cockcroft,
Gilbert & Walton with 0.6-MEV protons); C (?) (Henderson et al.,
with L5-MEV protons). Negative results by Henderson et al. with
L5-MEV protons on all but C among the elements listed above before
their names.

Bombardment by neutrons: F, Na, Mg, Al, Si, P, CI, Ti, V, Cr, Fe,
Cu, Zn, As, Se, Br, Zr, Ag, Sb, Te, I, Ba, La, U (Fermi); F, Mg, Al
(Dunning and Pegram).

Other Cases of Transmutation
It is not altogether safe to separate cases of "induced radioactivity"
from "other cases of transmutation," inasmuch as most of the latter
class have been observed under conditions where it was impossible to
tell whether or not there was a delay between collision and disruption,
and perhaps some of them belong in the former class. Of certain
transmutations one may say that if there is such a delay, the law of
conservation of momentum must be suspended for the duration
thereof, resuming its sway only at the moment of the disruption.
Nevertheless I should not wish to affirm that for the processes men-
tioned in this section or in the Second Part the delay is always literally
zero.

Early in this year was first achieved, at the Cavendish Laboratory
by Oliphant, Shire and Crowther, what had been the aim of many
physicists for over a decade : the separation of a metal, normally
consisting of more than a single isotope, into films each comprising
atoms of practically a single isotope only, and thick enough for physical
experiments. This was performed with lithium, and when protons
and alternatively deutons were projected against films of Li^ and
alternatively Li^, the four resulting sets of observations settled the
attributions of the various groups of fragments previously observed
when ordinary blocks of lithium had been bombarded. The origin of
the two long-range groups of paired alpha-particles described in the
Second Part was precisely as had been suspected: they proceed from
the interactions:

iHi + aLi^ = 22He^ + {T, - To); iH^ + ,U' = l^He' + {T, - To),
where (Ti — To) stands for the amount of energy transformed in each

400 BELL SYSTEM TECHNICAL JOURNAL

reaction from energy-of-rest-mass to kinetic energy, equal to about
17 MEV in the first case and to about 23 MEV in the second. The
continuous distribution of alpha-particles up to range 7.8 cm (Fig. 9
in the Second Part) is due to impacts of deutons against Li'^, and thus
may still be attributed to a transmutation in which three nuclei, — a
neutron and two alpha-particles — spring from the merger of a deuton
with a Li^ nucleus. Of the other attributions I shall presently speak.

The transmutations arising from the impact of deutons on deuterium
are in some ways unique. They are the first to be known in which
the two colliding particles are identical, both being H^ nuclei; one of
them appears to be much the most abundant yet observed, in the
sense that a given number of bombarding particles produces an
unprecedentedly great number of detectable fragments; each of them
results in the formation of a nucleus long sought but never certainly
detected till 1934.

The better-known of these reactions is described by the equation,

,H2 + ,W = iH" + iRi + {Tr - To). (1)

It is both somewhat amusing and somewhat annoying to realize that
this is not a transmutation at all in the formerly-proper sense of the
word, since there is no change of one element into another! the hydro-
gen isotope of mass-number 2 is changed into hydrogen isotopes of
mass-numbers 1 and 3 respectively; it will be desirable to enlarge the
scope of the term "transmutation" to cover cases like this one.
The H^ nuclei resulting from this reaction were vividly demonstrated
by Tuve and Hafstad when they projected deutons into divers gases
in an ionization-chamber — air, carbon dioxide, ordinary hydrogen,
and deuterium successively; there were no emerging protons (of range
superior to 3.5 cm, the minimum observable) from any of the three
first named, but from the last there was the "very large yield" of
one proton per several thousand impinging deutons. Another estimate
of yield has been supplied from the Cavendish school, by Oliphant
Harteck and Rutherford; theirs refers to impacts by deutons of
energy 0.1 MEV, a value considerably smaller than those of Tuve's
research; they find that the number of protons coming forth from a
thick layer of deuterium is of the order of a millionth of the number
of such deutons entering the layer. The estimates do not seem
incompatible, especially as the Cambridge people find the number of
fragments to be mounting very rapidly as the deuton-energy To
increases;*^ and they show that any possibility of a slight admixture

* The "thick layers" are fihns of certain compounds of hydrogen in which a
large proportion of the usual H' atoms have been replaced by H- atoms. The curve

CONTEMPORARY ADVANCES IN PHYSICS 401

of deuterium with any other substance must be very carefully con-
sidered and assessed, whenever that other substance is bombarded
with a beam containing deutons and it is observed that protons are
produced.

The range of the protons due to the foregoing reaction is about
14 cm when To is low — 0.1 MEV or thereabouts — and rises with Tq.
Translate its minimum value into the corresponding kinetic energy
(obtaining about 3 MEV); compute the momentum of the proton —
this, save for a minor correction due to the relatively small momentum
of the impinging deuton, should be opposite in direction and equal in
magnitude to the momentum of the other fragment of the transmu-
tation, the nucleus H^. Thence compute the kinetic energy of this
other fragment, and estimate thence its presumable range; owing to
our lack of experience with such particles the estimate may not be
very exact; Oliphant, Harteck and Rutherford arrive at the figure
1.74 cm. Now, the protons of 14-cm range of which I have been
speaking are not the only fragments to be observed when deutons
impinge on deuterium. There are also particles of a much less range;
these are equally numerous with the 14-cm protons, and expansion-
chamber photographs by Dee have shown that a track of the one
variety is likely to be paired with a track of the other, after the fashion
of the paired tracks due to the transmutations H + Li = 2He (Figs.
14 and 15, Second Part) ; and their range of about 1.6 cm. is taken by
the Cavendish people as being in substantial agreement with the
estimate aforesaid. It is this interlocking of concordant observations
which speaks so strongly for the Tightness of this description of the
reaction, and therefore for the existence of the hitherto-unknown
isotope H'' of hydrogen.

Meanwhile it has been discovered at Princeton that the new isotope
can be generated by maintaining a self-sustaining discharge in gaseous
deuterium: a way of achieving transmutation several times attempted
in past years, but never (so far as I know) with proved success. Out
from the discharge tube (where the voltage is 50,000 to 80,000) some
of the ionized atoms and molecules shoot through a hole in the cathode
into another and very large chamber filled with deuterium in which
they disperse themselves, thus having opportunities for transmutation
in both this chamber and the tube. A sample of the gas is afterwards

of number-of-fragments vs. To shows the pecuHar shape common to such curves
when obtained with thick layers, which suggests that as 7"o is raised the increase
in the number of transmutations is at first i)artly due to an increase in the [irobabiiity
of transmutation at an impact, but hiter entirely due to the fact that the faster
particles enter farther into the layer and have more ojiportiuiities of striking nuclei
before their energy is gone than do the slower (The Nucleus, Second Part, p. 141).
The theory of such curves has, however, never been worked out.

402 BELL SYSTEM TECHNICAL JOURNAL

extracted and is ionized in a separate chamber; the charge-to-mass
ratios of its ions are determined by an especial type of deflection-
apparatus. Search is made for ions having the charge-to-mass ratio
of a singly-ionized molecule of mass about 5, such as could be a
molecule H^H^. Such ions occur. To discover them, however, is not
the same thing as to prove the existence of H\ since so far as anyone
can tell from their charge-to-mass ratios (as measured with the
accuracy attainable in these experiments) these ions might have the
constitution H^H^H^ — there being some of the isotope H^ in the gas.
How to make such discriminations is one of the major problems in the
analysis of the ions found in gases. In this case it happens to be known
that in ordinary hydrogen, the ratio of the number of triatomic to that
of diatomic molecular ions is proportional to the density of the gas.
Now in these experiments, the ratio of the number of mass-5 ions to
the number of mass-4 ions is the sum of two terms, one proportional
to the gas-density and the other independent of it. The latter term
is taken as the measure of the amount of H-H^, therefore of H^, in the
gas. A like study made with deuterium none of which had been
exposed to the discharge indicated a very small amount of H^ about
one atom in two hundred thousand of H^; the discharge enhanced
this ratio fortyfold in an hour.

To return to the work at the Cavendish Laboratory: the lesser-
known of the two reactions which may occur when deutons meet is
probably described by the equation,

,W + iH2 = 2He'^ -f ,n' + {T, - To) (2)

and is a transmutation in the strictest sense of the word, helium as
well as neutrons ^ appearing out of hydrogen. I refer to it as lesser-
known, because although the neutrons have been observed the helium
nuclei have not been. This lack of evidence withholds a desirable
support from the equation, but does not contradict it; for on measuring
the momentum of the neutron, equating it to that of the hypothetical
He'^ nucleus and estimating the range of the latter, this range turns
out to be so small as to make detection difficult. We are not, however,
without other evidence for He^; when protons are projected against
lithium, particles of ranges 1.15 cm. and 0.68 cm. appear,^" and the
observations made with monisotopic films show that Li® is involved
in their origin: if we suppose

iRi + 3Li« = 2He^ + 2Ue + {Ti - To) (3)

'■• Harkins has suggested the name "neuton" for the element of which neutrons
are the ultimate particles.

1" Kirchner has lately observed an 0.9-cni. group.

CONTEMPORARY ADVANCES IN PHYSICS 403

the equation is supported by the facts that the ranges of the two
groups stand to one another in the ratio computed by assuming
equaUty of momenta, that particles of one are found to be paired
with particles of the other, and that they ionize about as much as
alpha-particles of equal range.

The rest-masses of the two new nuclei are estimated by putting,
in equations (1), (2) and (3\ the best available values for T^ (the
kinetic energy of the impinging deuton, that of the other H^ nucleus
being negligible) and Ti (the sum of the kinetic energies of both frag-
ments resulting from the reaction). The results are: for the rest-mass
of H\ 3.0151 from (1) ; for the rest-mass of He\ 3.0166 from (3). To
derive the latter from (2) is not so precise, the energy of neutrons
being harder to evaluate than that of charge-bearing particles; Oli-
phant, Harteck and Rutherford prefer to say merely that the result
is not incompatible with that from (3).^^

These are the fourth and fifth of the nuclei (counting the neutron
as one) in order of increasing mass. The departures of their masses
from the adjacent integer are abnormally great for light nuclei, and
their packing-fractions (First Part, p. 318) are the greatest yet known
excepting that for H-, and fall neatly by the upper branch of the curve
of packing-fraction vs. mass-number (Fig. 8 of the First Part). The
contrast between the packing-fractions 55 of Hc^ and 5 of He^ is
especially striking. The new nuclei are the first isobars to be dis-
covered of mass-number less than 40, and the first pair to be discovered
of which the masses are distinguishable.

Cockcroft and Walton have studied at length the fragments emerging
from lithium, boron and carbon bombarded by deutons. Lithium
supplies a group and boron a group of protons which may result from
the transformation of the lighter into the heavier isotope according
to the schemes,

,W + 3Li« = 3Li7 + iHi + {T, - To), (4)

xH2 + sBi" = 5B11 + iHi + {T, - To), (5)

but the two members of each equation (in which all the rest-masses
are known by deflection-experiments) do not agree very well. Carbon
supplies a group and boron two more groups of protons which cannot
be made to fit into such a scheme without postulating emission of
gamma-rays to achieve the balancing of masses — an emission for which,

1^ These results are computed by assuming that the values of the rest-masses of
H^, H-, He^ Li'', Li" and n^ given by Aston, Bainbridge and Chadwick are e.xact, and
that no additional fragment (such as a ganmia-ray photon) of appreciable energy is
emitted at the transmutation.

404 BELL SYSTEM TECHNICAL JOURNAL

it is true, independent evidence exists in the case of carbon. Boron
supplies a group of alpha-particles which may be due to the reaction,

iH2 + sBio = SzHe^ (6)

and which comprises the most energetic subatomic particles yet
known, those of the cosmic rays excepted (12.3 MEY, range 15 cm.).
Blocks of various heavier elements emit both alpha-particles and
protons, of which the amounts both relative and absolute vary tre-
mendously with heat-treatment, degassing, and other circumstances,
so that evidently they cannot altogether proceed from the element
constituting most of the block, and their origins furnish a severe
problem for research.

Electrical Wave Filters Employing Quartz Crystals as

Elements

By W. p. MASON

This paper discusses the use of piezo-electric crystals as elements in wave
filters and shows that very sharp selectivities can be obtained by employing
such elements. It is shown that by employing crystals and condensers
only, very narrow band filters result. By using coils and transformers in
conduction with crystals and condensers, wide-band-pass and high and low-
pass filters can be constructed having very sharp selectivities. The circuit
configurations employed are such that the coil dissipation has only the effect
of adding a constant loss to the filter characteristic, this loss being indepen-
dent of the frequency. Experimental curves are given showing the degree of
selection possible.

In the appendix, a study is made of the modes of motion of a perpen-
dicularly cut crystal, and it is shown that all the resonances measured can
be derived from the elastic constants and the density of the crystal. The
efTect of one mode of motion on another mode is shown to be governed by
the mutual elastic compliances of the crystal. By rotating the angle of cut
of the crystal, it is shown that some of the compliances can be made to
disappear and a crystal is obtained having practically a single resonant fre-
quency over a wide range of frequencies. Such a crystal is very advan-
tageous for filter purposes.

Introduction
■ FILTERS for communication systems must pass, without appreci-
-*- able amplitude distortion, waves with frequencies between certain
Hmits, and must attenuate adequately all waves with somewhat greater
or smaller frequencies. To do this efficiently, the change from the
filter loss in the transmission region, to that in the attenuation region,
must occur in a frequency band which is narrow compared to the use-
ful transmission band. At low frequencies, ordinary electrical coil
and condenser filters can perform this separation of frequencies well
because the percentage band widths (ratio of band width to the mean
frequency of the band) and the percentage separation ranges (ratio of
the frequency range required, in order that the filter shall change from
its pass region to its attenuated region, to the adjacent limiting fre-
quency of the pass band) are relatively large.

For higher frequency systems, such as radio systems, or high fre-
quency carrier current systems, the band widths remain essentially the
same, and hence the percentage band widths become much smaller.
Here separation by coil and condenser filters becomes wasteful of
frequency space. For these filters, owing to the relatively low react-
ance-resistance ratio in coils (this ratio is often designated by the
letter Q) the insertion loss cannot be made to increase faster than a

405

406 BELL SYSTEM TECHNICAL JOURNAL

certain percentage rate with frequency. Hence an abrupt frequency
discrimination cannot be obtained between the passed frequency range
and the attenuated frequency range. In present radio systems,
double or triple demodulation is often used to supplement the selectiv-
ity of filter circuits.

If, however, elements are employed which have large reactance-
resistance ratios, filters can be constructed which have small percent-
age bands and which attenuate in small percentage separation ranges.
Such high Q elements are generally obtainable only in mechanically
vibrating systems. Of these elements, probably the most easily used
is the piezo-electric crystal, for it possesses a natural driving mechan-
ism.

It is the purpose of this paper to describe work which has been done
in utilizing these crystals as elements in filters.^ Since the quartz
crystal appears to be the most advantageous piezo-electric crystal, all
of the work described in this paper is an application of this type of
crystal. The possibilities and limitations are discussed and experi-
mental data are given showing that these filters are realizable in a
practical form.

Piezo-electric Crystals and Their Equivalent Electrical

Circuits

When an electric force is applied to two plates adjacent to a piezo-
electric crystal, a mechanical force is exerted along certain directions
which deforms the crystal from its original shape. On the other hand
deformations in certain directions in the crystal produce a charge on
the electric plates. Hence the crystal is a system in which a mechanical
electrical coupling exists between the mechanical and electrical parts
of the system.

Quartz crystals, particularly when vibrating along their smallest
dimension, as they do for high frequency oscillators, have a large num-
ber of resonances which do not differ much in frequency from the prin-
cipal resonance. While this is no great disadvantage for an oscillator,
since an oscillator can pick out the strongest resonance and utilize it
only, the large number of resonances is a great disadvantage when using

1 The development of ideas in the direction of employing crystals as elements of
selecting circuits dates back to Cady who in patent — Re. 17,358 issued July 2, 1929,
original filed January 28, 1920 — showed various types of tuned circuits of which
crystals formed a part. Subsequently Espenschied in patent 1,795,204, issued
March 3, 1931, filed January 3, 1927 — patented broadly the use of crystals as ele-
ments of true filter structures. More recently a patent of the writer's — 1,921,035
issued August 8, 1933, filed Sept. 30, 1931 — describes the use of crystals in lattice
structures, and this patent, together with several others pending, forms the basis
for the filters discussed in this paper. It is only within the last few years that filter
structures including crystal elements have been practically realized.

ELECTRICAL WAVE FILTERS

407

the crystal to select currents over a wide band of frequencies and to
reject currents whose frequencies lie outside this pass region. Hence
it is advantageous for filter uses to obtain a crystal which has sub-
stantially a single prominent resonance over a wide range of fre-
quencies. Such a vibrating element can usually be obtained only by
making the dimension along which the crystal is vibrating, large
compared to the other dimensions, and this fact determines the best
cut of crystal to use.

Two principal types of orientations have usually been employed in
cutting quartz crystals. The first type is the so-called Curie or per-
pendicular cut in which the crystal is so cut that its major surfaces are
parallel to the optical axis and perpendicular to an electrical axis.
Such a crystal is shown by Fig. 1. The second type of cut is the parallel

OPTICAL AXIS=Z

MECHANICAL AX(S = Y

-ELECTRICAL AXIS=X

Fig. I — Orientation of a Curie or perpendicular cut with respect to native crystal.

or 30-degree cut in which the major surfaces of the crystal plate are
parallel to both the optical and electrical axes. Since this cut results
in a crystal vibrating along its smallest dimension, it is not of much
interest for filter uses.

When using a crystal as part of an electrical system, it is desirable

408

BELL SYSTEM TECHNICAL JOURNAL

to know the constants of an electrical circuit which has the same im-
pedance characteristic as the crystal. If attention is confined to the
single prominent resonance, the electrical circuit representing the
crystal is as shown by Fig. 2. Some theoretical consideration has been

r^Hh

r-nnnp-n

--Hh

Ca-Co + C|

fR =

06=1; (co+c,;

Lb =

c-i^r

rv i-iC|

C|
C|Co

2Tr;L|C| " 2Tr^

Fig. 2 — Equivalent electrical circuit of piezo-electric crystal.

given to the electrical network representing perpendicularly cut crys-
tals by Cady,^ Van Dyke,^ Dye,^ Vigoureux ^ and others. Assuming a
quartz plate to have only plane wave motion, Vigoureux has investi-
gated the motion in such a plate, and has shown that there will be
resonances at odd harmonics of a particular frequency determined by
the length, and mechanical constants of the plate. In the neighbor-
hood of the fundamental resonance of the crystal, with the electrical
plates placed on the crystal surfaces, he finds the equivalent circuit
shown by Fig. 2A, the elements of which in practical units have the
following values:

Co = :; — ; — . . ^ ^ , . ^fi = capacitance m larads,

Ci =

Li =

Airle X 9 X IQii
loLSEdn''

TvHe X 9 X 10

lelmP X 9 X 10"

SloEHn''

Yl = capacitance in farads,
inductance in henries,

(1)

where /o, Im, h are respectively the lengths of the optical, mechanical,
and electrical axes in centimeters,

K = specific inductive capacitance = 4.55 for quartz,
E = Young's modulus = 7.85 X 10" for quartz,

dn = piezo-electric constant = 6.4 X 10~^ for quartz,

p = density = 2.654 for quartz.

2 W. G. Cady: Phys. Rev. XIX, p. 1 (1922); Proc. I. R. E. X, p. 83, (1922).

3 K. S. Van Dyke: Abstract 52, Phys. Rev., June, 1925; Proc. I. R. E., June, 1928.
' D. W. Dye: Proc. Phys. Soc, XXXVIII (5), pp. 399-453.

^ P. Vigoureux: Phil. Mag., Dec, 1928, pp. 1140-53.

ELECTRICAL WAVE FILTERS 409

Inserting these values, the element values in terms of the dimensions
become

Co = 0.402 X 10-12 ij^ll^^

Ci = 0.289 X 10-" hnUlle, (2)

Z/i = 1 18.2 Imle/lo-

From these values it is seen that there is a fixed ratio between these
two capacitances ^ of

r = Co/Ci = 140. (3)

As will be evident later, this ratio limits the possibilities of the use of
quartz crystals in filter circuits.

Experiments with quartz crystals, with electrodes contiguous to
the crystal surfaces and with the optical and electrical axes small
compared to the mechanical axis, show that these values are approxi-
mately correct. The value of Co checks the above theoretical value
quite closely. The value of Ci obtained by experiment is somewhat
larger than that given by equation (2) and the value of the inductance
somewhat smaller. The ratio of Co/Ci has been found as low as 115
to 1, but a value of 125 to 1 is about all that can be realized, when
account is taken of the distributed capacitance of the holder, connect-
ing wires, etc.

When either of the dimensions along the electrical or optical axes
becomes more than a small fraction of the dimension of the mechanical
axis, the plane wave equations given above no longer hold accurately.
This is due to the fact that a coupling exists between the motion along
the mechanical axis and other modes of motion. For an isotropic
body, one is familiar with the fact that when a bar is compressed or
stretched it tends to stretch or compress in directions perpendicular to
the principal direction of motion. This state of affairs may be de-
scribed by saying that the modes of motion are coupled. In a crystal
this same relation exists and in addition, due to its crystalline form, a
shearing motion is set up whose shearing plane is determined by the
mechanical and optical axes and whpse motion is parallel to the me-
chanical axis. In fact the shearing motion is more closely coupled to
the mechanical axis motion that is the extensional motion. As long
as the optical axis is small compared to the mechanical axis, this coup-
ling action manifests itself as a decreased stiffness along the mechanical
axis, but if a condition of resonance is approached for the motion along
the optical axis, the mode of motion is entirely changed. This effect is

^ In a paper contributed recently to the Inslilnte of Radio Engineers, it is shown
that this ratio limitation is a consequence of a fixed electro-mechanical coupling
between the electrical and mechanical systems of the crystal.

410

BELL SYSTEM TECHNICAL JOURNAL

analyzed in the appendix and is quantitatively explained in terms of
the elastic constants of the crystal. On the basis of this explanation,
an investigation is also given in the appendix, of crystals cut at differ-
ent orientations, and a crystal having many advantages for filter uses
is derived.

Some experimental data ^ have been taken for perpendicularly cut
crystals for various ratios of axes. Figure 3 shows the principal reso-

MECHANICAL AXES= 10 MILLIMETERS
ELECTRICAL AXES = 0.5 MILLIMETERS

3 4 5 6 7

DEPTH OF OPTICAL AXES IN MILLIMETERS

Fig. 3 — Principal resonant frequency of a perpendicularly cut crystal as a function of

the width of the crystal.

nant frequency (the frequency for which the electrical impedance is a
minimum) for a series of crystals whose mechanical axes are all 10
millimeters, whose electrical axes are 0.5 millimeter, and whose optical
axes vary from 1 to 10 millimeters. As will be observed, increasing the
length of the optical axis in general lowers the resonant frequency.
The discontinuity in the curve for the ratio hjlm = -23 is discussed in
detail in the appendix.

^ The experimental data shown by Figs. 3 and 4 have been taken by Mr. C. A.
Bieling while the temperature coefficient curve of Fig. 5 was measured by Mr. S. C.
Hight.

ELECTRICAL WAVE FILTERS

411

The solid curve of Fig. 4 shows a measurement of the ratio, r, of the
capacitances in the simple representation of the crystal shown by Fig.
2A. This ratio is measured by determining the resonant and anti-

0 170

O 160

MECHANICAL AXES = 10 MILLIMETERS
ELECTRICAL AXES = 0.5 MILLIMETERS

11
1
1

ll
1
1

1
1
1
1

1
1

[l
1
1

ll
1

//
//
//
/ /

)

^"^

3 4 5 6 7

DEPTH OF OPTICAL AXES IN MILLIMETERS

Fig. 4 — Ratio of capacitances of a perpendicularly cut crystal.

resonant frequencies of the crystal, r is related to these by the
formula

fAVfR' = 1 + 1/r,

(4)

where /a is the anti-resonant frequency and/ij the resonant frequency.
Figure 5 shows a measurement of the temperature coefficient of the
resonant frequency for the same set of crystals. It will be noted that
as the optical axis increases in depth, the temperature coefficient
increases and that crystals with smaller dimensions along the optical
axis in general have much smaller coefficients. Increasing the thick-
ness along the electrical axis has the efifect of decreasing the tempera-

412

BELL SYSTEM TECHNICAL JOURNAL

ture coefficient and in fact for certain ratios of the three axes the coeffi-
cient approaches zero.

z yj

- > 40

u. z
W o

uj -

cr S

b ^

< UJ

cr a

CL 1/1

lij □:

H <

MECHANICAL AXES =10 MILLIMETERS
ELECTRICAL AXES =0.5 MILLIMETERS

■^

0 123456789 la

DEPTH OF OPTICAL AXES IN MILLIMETERS

Fig. 5 — Temperature coefficient of a perpendicularly cut crystal.

When the crystals are used in filters, two quantities are usually-
specified, the resonant frequency of the crystal and the capacitance of
the series condenser. The shunt capacitance of the crystal is usually
incorporated with an electrical capacitance which is specified by other
considerations. The resonant frequency is determined principally by
the mechanical axis length. The capacitance of the series condenser
is determined by the ratio of the area to the thickness or by Ulmlle.
The third condition is given by the fact that the length of the optical
axis should be kept as small as possible in order that any subsidiary
resonances shall be as far from the principle resonance as possible.
The curves of Figs. 3 and 4 and the fact that the resonant frequency of
a given shaped crystal varies inversely as the length, can be used to
determine the dimensions of the crystal. It is obvious that the
crystal should not be used in the region 0.2 < Ujlm < 0.3 on account of
the two prominent resonant frequencies.

Use of Crystals and Condensers as Filter Elements
Considering crystals as representable by the simple electrical
circuit shown on Fig. 2A , these circuits can be utilized as elements in
electrical networks. They may, of course, be used in a network em-
ploying any kinds of electrical elements. Since, however, their Q is
high, it would be advantageous not to employ any electrical elements
which do not also have a high Q, in order that the dissipation intro-
duced by these elements may not be a matter of consideration. The

ELECTRICAL WAVE FILTERS

413

(2's of the best electrical condensers may be as high as 10,000, which
is of the same order as the crystal Q, and hence such elements can be
employed advantageously with crystals. It is the purpose of this
section to discuss the possibilities and limitations of filter sections em-
ploying crystals and condensers only.

The simplest types of filter sections are the ladder type networks
illustrated by Fig. 6. If crystals and condensers only are employed in

U/^ 2C| 1-1^201

ELECTRICAL

STRUCTURE

X2 ' "/Z^'-\

2C2

2C2 =L 2C2

1 C4

Ll^ 2C| 1-1/2 2C|

2C2l-3g Xc4 2C2

PHYSICAL
STRUCTURE

Ca =i=C4 Ca

2C2

I — I -*— f

QSIZD 4=c

REACTANCE

CURVES FOR

EACH ARM

ATTENUATION

CHARACTERISTIC

FREQUENCY

MID SERIES
ITERATIVE
IMPEDANCE

• |FRE

_^^ Iquency]

y h If*"/-" °°

FREQUENCY

yfl % ,-^

-R^.

■FREQUENCY

fl/ f2 f0O2|,^'

MID SHUNT
ITERATIVE
IMPEDANCE

y¥y

FREQUENCY

i.ool' f2 /

I FREQUENCY

' fl /

hJ\

fooi/| 1 FREQUENCY

fl f2l ,'

Fig. 6 — Ladder networks employing crystals and condensers.

this type network, there are only three types of single band sections
possible, all being band-pass filters. Figure 6 shows these sections, the
impedance characteristic of each arm, the attenuation characteristics
of these networks considered as filters, and their iterative impedances.

414 BELL SYSTEM TECHNICAL JOURNAL

These attenuation characteristics and their limitations are at once
found from a consideration of the impedance frequency curves for each
arm shown by Fig. 6C. For a ladder type network it is well known ^
that a pass band will exist when

OgA=_i, (5)

where Zi is the impedance of the series arm and Z^ the impedance of
the shunt arm. Hence, considering the first filter of Fig. 6, it is ob-
vious that the lower cut-off /ci will come at the resonant frequency of
the crystal. The upper cut-off /c2 will come some place between the
resonant and anti-resonant frequency, the exact position depending
on the amount of capacitance in shunt. The anti-resonant frequency
will be a point of infinite attenuation since the filter will have an in-
finite series impedance at this frequency.

With the restriction on the ratio of capacitances of the crystal
noted in the previous section, it is easily shown that the ratio of the
anti-resonant frequency to the resonant frequency is fixed and is about
1.004. Hence, we see that the ratio of /oo to /ci can be at most 0.4
per cent. The band width must be less than this since /C2 must come
between /oo and /ci. A similar limitation occurs for the second filter
of this figure, for which case the separation of /<» and/c2 is at most 0.4
per cent. For filter number 3, a somewhat larger frequency separa-
tion between the points of infinite attenuation results, it being at most
0.8 per cent. The addition of any electrical capacitance in series or
shunt with any of the crystals results in a narrowing of the band width.

It is seen then that there are two limitations in the types of filters
obtainable with crystals and condensers in ladder sections. One,
there is a limitation on the position of the peak frequencies and two,
there is a limitation for the band width of the filters.

By employing the more general lattice type of filter section shown
on Fig. 7, the first of these limitations can be removed. By means of
this type of section it is possible to locate the attenuation peak fre-
quencies at any position with respect to the pass band, but the pass
band is limited in width to at most 0.8 per cent.

For a lattice filter a pass band exists when the impedances of the
two arms are related by the expression ^

0^1-^^--, (6)

* See, for example, page 190 in book by K. S. Johnson, "Transmission Circuits for
Telephonic Communications."

9 "Physical Theory of the Electric Wave Filter," G. A. Campbell, B. S. T, J.,
November, 1922.

ELECTRICAL WAVE FILTERS

415

where Z\ is the impedance of the series arm (either 1, 2 or 3, 4 of Fig.
1A) and Z^ the impedance of the lattice arm (either 1, 3 or 2, 4 of
Fig. lA). Hence, if one pair of branches has a reactance whose sign

A-ELECTRICAL STRUCTURE

B- PHYSICAL STRUCTURE

C-CHARACTERISTIC
ATTENUATION

■CHARACTERISTIC
IMPEDANCE

REACTANCE CURVES
FOR EACH ARM

FREQUENCY

/^1

FREQUENCY

Fig. 7 — Lattice network employing crystals and condensers.

is opposite to that of the other pair, a pass band exists, while if they are
the same sign an attenuated band exists. Since the lattice is in the
form of a Wheatstone bridge an infinite attenuation exists when the
bridge is balanced, which occurs when both pairs of arms have the same
impedance.

Let us consider a lattice filter with a crystal in each arm as shown
by Fig. IB. The crystals form two pairs of identical crystals, two
alike in the series arms and two alike in the lattice arms. In order that
a single band shall result it is necessary that the anti-resonant frequency
of one arm coincide with the resonant frequency of the other as shown
by Fig. IE. It is obvious that the band width will be twice the width
of the resonant region of the crystal or at most 0.8 per cent. Since the
attenuation peaks occur when the two arms have the same impedance,
they may be placed in any desired position by varying the impedance
of one set of crystals with respect to the other. If crystals alone are
used, these peaks will be symmetrical with respect to the pass band,
but if in addition condensers are used with these crystals, the peaks may
be made to occur dissymmetrically. In fact they may be made to
occur so that both are on one side of the pass band. A narrower band
results when capacitances are used in addition to crystals since the
ratio of capacitances becomes larger. This may be utilized to control
the width of the pass band to given any value less than 0.8 per cent.

416 BELL SYSTEM TECHNICAL JOURNAL.

The use of more crystals than four, in any network configuration
employing only quartz crystals and condensers can be shown to result
in no wider bands than 0.8 per cent, although higher losses can be ob-
tained by the use of more crystals. Hence by the use of quartz crys-
tals and condensers alone, a limitation in band width to 0.8 per cent
is a necessary consequence of the fixed ratio of capacitances Co/Ci of
equation (3),

Filter Sections Employing Crystals, Condensers and Coils
As was pointed out in the last section, filters employing only
crystals and condensers are limited to band pass sections whose band
widths do not exceed about 0.8 per cent. This band width is too nar-
row for a good many applications and hence it is desirable to obtain a
filter section allowing wider bands while still maintaining the essential
advantages resulting from the use of sharply resonant crystals. Such
filters can be obtained only by the use of inductance coils as elements.
Since the ratio of reactance to resistance of the best coils mounted in a
reasonable space does not exceed 400, attention must be given to the
effect of the dissipation.

The effect of dissipation in a filter is two-fold. It may add a con-
stant loss to the insertion loss characteristic of the filter, and it may
cause a loss varying with frequency in the transmitting band of the
filter. The second effect is much more serious for most systems since
an additive loss can be overcome by the use of vacuum tube amplifiers
whereas the second effect limits the slope of the insertion loss frequency
curve. Hence, if the dissipation in the coils needed to widen the band
of the filter has only the effect of increasing the loss equally in the
transmitting band and the attenuating band of the filter, a satisfactory
result is obtained. The question is to find what configuration the
coils must be placed in with respect to the crystals and condensers in
order that their dissipation will not cause a loss varying appreciably
with frequency.

Not every configuration will give this result, as is shown by the fol-
lowing example. The equivalent circuit of the crystal shown by Fig. 2A
can be transformed into the form shown by Fig. 8^ where the ratio
Ci/Co = 125. This gives the same reactance curve as before, limited
to a width of 0.4 per cent. Now suppose that we add an electrical
anti-resonant circuit in series with the crystal — Fig. 85 — resonating
at the same frequency and having the same constants as the anti-
resonant network representing the crystal. If this circuit were dissipa-
tionless we could combine the two resonant circuits into one with twice
the inductance and half the capacitance of that for the crystal alone

ELECTRICAL WAVE FILTERS

417

and hence the capacitance ratio would be 1/2 (125) or 62.5. The band
width possible would then be twice that of the crystal alone. However,
when the effect of dissipation is considered it is found that not much
has been gained by employing the anti-resonant circuit. For the re-

/TTf\

Hh' - -'

rHnnrs

= 125

ABC
Fig. 8 — Use of an anti-resonant circuit to broaden the resonance region of a crystal.

sistance, at resonance of the crystal and electrical circuit combination,
will be the resistance of the electrical resonant circuit since that of the
crystal is small compared to the electrical element. Hence we have
doubled the impedance of the anti-resonant circuit and have the re-
sistance of the electrical circuit. Hence the ratio {Q) of reactance to
resistance of the anti-resonant circuit is double that of the electrical
element alone. Even this Q, however, is insufficient to make a narrow
band filter whose band width is 1.6 per cent (twice that possible with
a crystal alone) and hence no useful purpose is served by combining a
crystal with an electrical anti-resonant circuit.

I — mn — I

2
Ro-R|

I — WW —

FILTER
SECTION

\A/W^ 1

Ro-R|-

— WW — '

>RoR| t.

|R|

FILTER

R|!

> <

S RoR|<

|R|-R0 1

SECTION

> R|-Ro<
>. <

j» <

A B

Fig. 9 — Circuit showing resistances on the ends of filter sections.

Suppose, however, that all of the dissipation of the filter section be
concentrated at the ends of the sections, either in series or in parallel
with the filter as shown on Fig. 9. Then provided these resistances are
within certain limits, they can be incorporated in the terminal resist-
ances of the filter by making these resistances either smaller or larger
for series or shunt filter resistances respectively. Between sections the
resistances on the ends of the filter can be incorporated with other

418

BELL SYSTEM TECHNICAL JOURNAL

resistances in such a way as to make a constant resistance attenuator of
essentially the same impedance as the filter. For a series coil, this
can be done by putting a shunt resistance between sections, while for a
shunt coil it can be done by putting series resistances between sections.
If this is done the whole effect of the dissipation is to add a constant
loss to the dissipationless filter characteristic, this loss being independ-
ent of the frequency.

Since the lattice type network provides the most general type of
filter network, attention will first be directed to this type of section
employing inductances. It is easily proved that if any impedance is in
series with both sides of a lattice network, as shown by Fig. 10^, then

A c

-*^AAAA/ ^AAAA

•-AAAA/—

»^ \NV\f—^

■ \/ ■

Fig. 10 — Two network equivalences.

this is equivalent to placing this impedance in series with each arm of
the lattice network as shown. Similarly, if a given impedance shunts
the two ends of a lattice network, as shown by Fig. lOB, a lattice net-
work equivalent to this is obtained by placing the impedance in shunt
with all arms of the lattice. We are then led to consider a lattice net-
work which contains coils either in series or in shunt with the arms of a
lattice network, these arms containing only crystals and condensers,
since the dissipation will then be effectively either in series or in shunt
with the lattice network section.

If an inductance is added in series with a crystal the resulting re-

ELECTRICAL WAVE FILTERS

419

actance is shown by the full line of Fig. 1 1 ; the dotted lines show the
reactance curves for the individual elements. It is evident that the
resonant frequency of the crystal is lowered, the anti-resonant, point
remains the same, and an additional resonance is added at a frequency

Fig. 11 — ^Impedance characteristic of crystal and coil in series.

above the anti-resonant frequency. For a crystal whose ratio of ca-
pacitances r is about 125 it is easily shown by calculation that if the
resonances are evenly spaced on either side of the anti-resonant fre-
quency the percentage frequency separation between the upper reso-
nance and the lower resonance is in the order of 9 per cent.

Suppose now that this element is placed in the series arm of a lattice
network and another element of similar character is placed in the lat-
tice arm, the second element having its lowest resonance coincide
with the anti-resonance of the first element, and having the anti-
resonance of the second element coincide with the highest resonance of
the first element. This condition is shown by Fig. 12 C This network
will produce a band-pass filter whose band extends from the lowest
resonance of the series arm to the highest resonance of the lattice arm,
a total percentage frequency band width of 13.5 per cent. By design-
ing the impedances correctly the impedances of the two arms can be
made to coincide three times so that there is a possibility of three
attenuation peaks due to this section as shown by Fig. \2D. The loss
introduced by the filter is equivalent to that introduced by three simple
band-pass sections. Ordinarily the coils in the two arms are made
equal so that their resistances are equal and for this case one of the
peaks occurs at an infinite frequency. Since the resistances are
equal, then by the theorem illustrated by Fig. 10^ these resistances
can be brought out on the ends and incorporated with the terminal

420

BELL SYSTEM TECHNICAL JOURNAL

resistances, with the result that the dissipation of the coils needed to
broaden the band has only the effect of adding a constant loss to the
filter characteristic, this loss being independent of the frequency.

ELECTRICAL

STRUCTURE

PHYSICAL

STRUCTURE

REACTANCE

CURVES FOR

EACH ARM

ATTENUATION
CHARACTERISTIC
D

ITERATIVE
IMPEDANCE

BAND- PASS FILTER

rw^^'

FREQUENCY

y FREQUENCY

Fig. 12— Lattice network band-pass filter employing series coils.

To vary the width of the band below the 13.5 per cent band ob-
tained with crystals only, added capacitances can be placed in parallel
with the crystals increasing the ratio r. This results in a smaller separa-
tion in the resonant frequencies and hence a narrower band width.
By this means the band width can be decreased indefinitely, although
the dissipation caused by the coils introduces large losses for band
widths much less than 1/2 per cent. By this means, however, it is
possible to obtain band widths down to the widths which can be real-
ized with crystals alone. On the upper side electrical filters can be
built whose widths are as small as 13.5 per cent, hence this method fills

ELECTRICAL WAVE FILTERS

421

in a range not practical with electrical filters, or with crystals alone.
Another important characteristic of the filter is its iterative im-
pedance. For a lattice filter this is given by ^

Zi = VZ1Z2,

where Zi is the impedance of the series arm and Z2 that of the lattice
arm. For a dissipationless filter, this is shown by Fig". 12£, as can be
easily verified by a consideration of the reactance curves of Fig. 12 C

ELECTRICAL

STRUCTURE

PHYSICAL

STRUCTURE

REACTANCE

CURVES FOR

EACH ARM

ATTENUATION

CHARACTERISTIC

ITERATIVE
IMPEDANCE

BAND -PASS FILTER

^^r'

/ 1 FREQUENCY

/f"^

FREQUENCY

fi fal

I /

Fig. IS^Lattice network band-pass filter employing parallel coils.

This type of filter results in a relatively low impedance, for example
about 600 ohms for a filter whose mid-band frequency is 64 kilocycles
and whose band width is that shown on Fig. 19. Since the band width
is decreased by adding more capacitance, it is evident that smaller

422

BELL SYSTEM TECHNICAL JOURNAL

percentage band width filters will have lower impedances than the
wider ones. For example, the filter whose characteristic is shown by-
Fig. 20, has an iterative impedance of 25 ohms.

It is evident that a still wider band can be obtained with the sec-
tion discussed above by making the two resonances of Fig. 11 dissym-
metrical. If the lower one is brought in closer to the anti-resonant
frequency the top one extends farther out in such a manner that the
total percentage frequency separation is greater than 9 per cent. If
one element of this type is combined with one whose lower resonance is
brought farther away from the anti-resonance than is the upper reso-
nance, a filter whose pass band is greater than 13.5 per cent is readily
obtained. On the other hand as the band is widened by this means,
the cross-over points of the impedances of the two arms are of necessity
brought very close to the cut-off frequencies, so that such a filter would
introduce most of its loss very close to the cut-off frequencies. This
type of characteristic might be useful in supplementing the loss charac-
teristic possible with electrical elements, but by itself would not pro-
duce a very useful result.

We have so far discussed the characteristics which can be obtained
by placing coils in series with crystals. An equally useful result is
obtained by placing coils in shunt with crystals as shown by Fig. 135.
This arrangement results in a band-pass filter capable of giving the
same band width as the first type discussed above. The only difference

Fig. 14 — Band-pass filter used between vacuum tubes.

occurs in the iterative impedance which will be as shown by Fig. \ZE.
For narrow band widths this type of filter has a very high iterative
impedance. For example, for a one per cent band width, using ordin-
ary sized coils and crystals, the iterative impedance may be as high as
400,000 ohms. Such filters can be used advantageously in coupling
together high impedance screen gird tubes without the use of trans-
formers. One such circuit is shown schematically by Fig. 14.

Filters made by using either series or shunt coils in conjunction

ELECTRICAL WAVE FILTERS

423

with condensers and crystals make very acceptable band-pass filters
capable of moderate band widths. It is often desirable to obtain low
and high-pass filters having a very sharp selectivity. The filter of Fig.
12 can be modified to give a high-pass characteristic by leaving out
the coils in the series or lattice arms of the network. However, it will
be found that the cross-over points in the impedance curve of necessity
come very close to the pass band and hence no appreciable loss can be
maintained at frequencies remote from the pass band. A broader and
more useful characteristic is obtained by using a transformer having
a preassigned coefficient of coupling, in conjunction with crystals and
condensers, as the element for broadening the separation of resonances.
Such an element is shown by Fig. 15^. As is well known, a trans-

fa A

FREQUENCV

Fig. 15 — Impedance characteristic of a transformer, condenser, and crystal.

former with a specified coupling can be replaced by a T network of
three inductances as shown by Fig. \SB. The impedance character-
istic, as shown by Fig. 15 C, has two anti-resonant frequencies /i and /a,
and two resonant frequencies f^ and fi.

Suppose now that an element of this type is placed in one arm of the
lattice and a similar element having a condenser in series with it is
placed in the other arm as shown by Fig. \6A. If the elements are so
proportioned that the anti-resonances of one arm coincide with the
resonances of the other arm and vice versa, as shown by Fig. \6B, the
impedances of the two arms are of opposite sign till the last resonance.
Hence, a low pass filter results. It is possible to make the two im-
pedance curves cross five times, so that an attenuation corresponding
to five simple sections of low-pass filter results. Other arrangements of
the resonances are also possible and are advantageous for special
purposes. For example, as shown by Fig. 16C we can make the last
resonance and anti-resonance of both arms coincide, and the other
resonances of one arm coincide with the anti-resonances of the second
arm. This arrangement results in a low-pass filter having an attenua-
tion corresponding to three simple low-pass filter sections and an
impedance which can be made nearly constant to a frequency very near
the cut-off frequency. This is advantageous for obtaining a filter with

424

BELL SYSTEM TECHNICAL JOURNAL

a sharp cut-off, for otherwise the mismatch of impedance near the
cut-off frequency causes large reflection losses which prevent the
possibility of obtaining a sharp discrimination.

r^^H

20 ^

" /
" /

1 1 /

/ 1 /
/ /

'1/

/ /

/ ' FREQUENCY

/
/
/
/

1
1

/I /
/ 1 '
/ 1 '

\ 1'
1

/ I'

/ 1'

Fig. 16 — Lattice network low-pass filter employing transformers, condensers, and

crystals.

The effect of dissipation in the transformer on the loss characteristic
is not so easy to analyze in this case as in the case of a series coil. The
effect can be obtained approximately as follows. Of the three coils of
Fig. \5B representing the transformer, the shunt coil has the least
dissipation since no copper losses are included in this coil. For an
air core coil, the Q of this shunt coil becomes very high and its dissipa-
tion can be neglected. The resistance of the primary winding can be
incorporated in the terminal resistance as in the series coil type of
filter and hence will cause only an added loss. The resistance of the
secondary will be in series with the crystal and condenser, and for a
reasonably good coil is of the same order of magnitude as the crystal
resistance at resonance. Hence its effect will be much the same as
cutting the Q of the crystal in half, so that instead of a crystal whose
Q is 10,000, we use one whose Q is 5000 and a dissipationless coil.
We see then that the Q of the crystal is still the most important factor
in determining the sharpness of cut-off in the filter as in the previous

ELECTRICAL WAVE FILTERS

425

ones described, and hence a very sharp selectivity can be obtained with
this circuit. It is possible to save elements in this filter by using two
primaries for each coil, putting one primary in one series or lattice arm
and the other in the corresponding series or lattice arms as shown by
Fig. \6D. Only half the number of elements per section are required.

By replacing the series condenser of the series arm of Fig. \6A by a
parallel condenser, it is possible to change the filter fi-om a low-pass
to a high-pass filter. Condensers in series, or in parallel with both
arms result in wide band-pass filters. It is possible to obtain a wider
pass band with this type of filter than with the single coil type since the
resonances will be spread over a wider range of frequencies.

In a good many cases it is desirable to have unbalanced filter sec-
tions rather than the balanced type which results from the use of a
lattice network. This is particularly true for high impedance circuits
for use with vacuum tubes. Since the lattice type section is the most
general type, it gives the most general characteristics obtainable.
The filter sections described here can in some cases be reduced to un-
balanced bridge T sections by well known network transformations,
with, however, more restrictions on the type of attenuation character-
istics physically obtainable.

A very simple bridge T network, which is equivalent to a lattice
network of the kind shown on Fig. 13, with two crystals replaced by
condensers, is shown on Fig. 17. This section employs mutual induct-

Fig. 17 — Single crystal bridge T band-pass filter.

ance, and the resistance ^^ shown is necessary in order to balance the
arms of the equivalent lattice. This type of network is able to repro-
duce some of the characteristics of the lattice filter, but is not so general
and is, moreover, affected by the dissipation of the coil to a larger extent
than the equivalent lattice.

'" The use of this resistance was suggested by Mr. S. Darlington and practically
all the work of developing this filter has been done by Mr. R. A. Sykes.

426

BELL SYSTEM TECHNICAL JOURNAL

Experimental Results

A number of filters have been constructed, during the past four
years, which employ quartz crystals as elements. Figure 18 shows the
measured insertion loss characteristic of a narrow band filter ^^ employ-

70
60

1
/

1
1

\
\

50
40
30
20
10
0

149.0 149.2 149.4 149.6 149.8 150.0 150.2 150.4 150.6 150.8 151.0
FREQUENCY IN KILOCYCLES PER SECOND

Fig. 18 — Measured insertion loss characteristic of a narrow band-pass filter.

ing only crystals and condensers. This filter employs two sections of
filter No. 3 of Fig. 6. It will be noted that in spite of the very narrow
band width, the insertion loss in the transmitted band is quite small.
A number of the broader band filters employing coils as well as
condensers and crystals have also been constructed. The frequency
range so far developed extends from 36 kilocycles to 1200 kilocycles.
Figure 19 shows the insertion loss characteristic of a band-pass filter
whose mid-frequency is 64 kilocycles and whose band width is 2500
cycles. The insertion loss rises to 75 db, 1500 cycles on either side of
the pass region. This filter was constructed from two sections of the
band-pass type described in Fig. 12. A similar insertion loss character-
istic, but shifted to a higher frequency, is shown by Fig. 20. The
insertion loss in the center of the band for this higher frequency filter
is considerably larger due to the smaller percentage band width. It is
interesting to note that practically all of this loss is due to the dissipa-
tion introduced by the coils. The useful percentage band width is
about one-half per cent and the filter reaches its maximum attenuation

" The filters whose characteristics are shown on Figs. 18 and 21 were designed and
constructed by Messrs. C. E. Lane and W. G. Laskey. The author wishes to call
attention to the fact that they and others associated with them in the Laboratories
have made considerable progress in connection with the practical difficulties en-
countered in the design and construction of these filters such as working out the
high precision element adjustment methods required, in methods of mounting, and
in shielding methods.

ELECTRICAL WAVE FILTERS

427

LlJ

'^ 60
a

- 50
<f)
in
o

-1 40

z
o

H 30

(r
u
m
z 20

58 59

61 62 63 64 65 66 67 68 69

FREauENCY IN KILOCYCLES PER SECOND

Fig. 19 — Measured insertion loss characteristic of a band-pass filter.

a. 40

\A ^___

"v^

^^/ t

^ 5

496 497 498 499 500

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 20 — Measured insertion loss characteristic of a band-pass filter at a high

frequency.

428

BELL SYSTEM TECHNICAL JOURNAL

in less than one-fourth per cent frequency range on either side of the
pass band.

Figure 21 shows the insertion loss characteristics of a filter employed

55
50

|45

140

;35

,30

I 25

; 20

; 15

—

Fij

144 146 148 150 152 154 156 158 160 162

FREQUENCY IN KILOCYCLES PER SECOND

;. 21 — Measured insertion loss characteristic of a band-pass filter used in a single
side band radio receiver.

in an experimental radio system for separating the two sidebands of a
channel at a high frequency. Here the separation- is effected in about
0.15 per cent frequency range. With the best electrical filters the
frequency space required for such a separation is about 1.5 per cent.
Figure 22 shows the insertion loss characteristic of a high-pass filter

100

210

:30 250 270 290 310 330 350 370

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 22 — Measured insertion loss characteristic of a high-pass filter.

ELECTRICAL WAVE FILTERS

429

constructed by using the circuit of Fig. 16Z), modified by using a paral-
lel condenser rather than a series condenser. The filter obtains a
65 db discrimination in less than a 0.12 per cent frequency separation.
Figure 23 shows a characteristic obtained by employing a filter of

1193 1197 1201 1205 1209

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 23 — Measured insertion loss characteristic of a single crystal bridge T filter.

the type shown by Fig. 17, together with a screen grid vacuum tube.
The result is plotted as the gain of the circuit since this gives the most
significant result for this type of circuit.

APPENDIX V,

The Modes of Vibration of a Perpendicularly Cut Quartz

Crystal

Introduction
Quartz crystals have been cut into two principal types of orienta-
tions with respect to the natural crystal faces. The first type is the so-
called Curie or perpendicular cut in which the crystal is so cut that its
major surfaces are perpendicular to an electrical axis and parallel to the
optical axis. Figure 1 shows such a cut. The second type is the so-
called parallel or 30-degree cut in which the major surfaces are parallel

430 BELL SYSTEM TECHNICAL JOURNAL

to both the optical and electrical axes. In this appendix a study is
made of the modes of motion of a perpendicularly cut crystal. The
effect has been studied of rotating the direction of the principal axis
while still maintaining the principal surfaces perpendicular to the
electrical axis. Such a crystal is designated as a perpendicularly cut
crystal with an angle of rotation 6.

The perpendicularly cut crystal has received considerable theoret-
ical and experimental consideration especially from Cady,^ Van Dyke,^
Dye ^ and Vigoreux.^ They have assumed that the crystal has a plane
wave vibration, and have calculated the frequencies of resonance in
terms of the elastic constants and the density of the crystal, and have
derived equivalent electrical networks for giving their electrical im-
pedance. Such representations indicate that there should be one
resonance for the crystal, the frequency of which is inversely propor-
tional to the length and independent of the width of the crystal. As
long as the length of the mechanical axis is large compared to that of
any other axis, this prediction agrees with the experiment, but when
the length of the other axes become comparable with that of the
mechanical axis, the prediction is no longer fulfilled by experiment.
It has long been recognized that this deviation is due to the failure of
the plane wave assumption. Rayleigh ^^ has given a correction for
taking account of lateral motion, which is applicable to an isotropic
medium. In a crystal, shear vibrations may be set up as well and
for this case Rayleigh 's correction can only be regarded as qualitative.
Also if the other sets of resonance frequencies are to be investigated,
account must be taken of the resonances of the other modes of vibra-
tion, and their reaction on the mode to be studied.

In this appendix experimental results have been obtained showing
the frequencies of resonance found in perpendicularly cut crystals of
various shapes and orientations. These frequencies are correlated
with the elastic constants of the crystal and are shown to be com-
pletely accounted for by them. A coupled circuit representation is
developed which is capable of predicting the main features of the
principal vibration, including the change of frequency with the shape
and orientation of the crystal, and the temperature-»Goefficient curves.

Experimental Determination of the Resonant Frequencies
In order to investigate the modes of motion in a perpendicularly
cut crystal in which the main axis coincides with the mechanical axis
of the crystal, a set of measurements has been made on crystals whose

2.3.4.5LoC_ Cit.

^^ Rayleigh, "Theory of Sound," Vol. I, Chapter VII, page 252.

ELECTRICAL WAVE FILTERS

431

mechanical axes are all 1.00 centimeter long, whose electrical axes are
very thin, being 0.05 centimeter, and whose optical axes vary in
dimension from 0.1 centimeter to 1.00 centimeter. In order to eliminate
the effect of a series capacitance due to an air gap, the crystals were
plated with a very thin coat of platinum. The effect of an added shunt
capacitance in parallel with the crystal, due to the electrode capaci-
tances, was practically eliminated by running the crystal electrodes in
an outer grounded conductor as shown in Fig. 24, which shows the

100 OHM
RESISTANCES

OSCILLATOR

OUTER CONDUCTOR
V

INNER
CONDUCTOR

-^±=- CRYSTAL

INSULATED
BEARINGS

INSULATOR

DETECTOR

OSCILLATOR

DETECTOR

Fig. 24 — Measuring circuit used to measure the resonances of a crystal.

measuring circuit. Contact to the crystal plating is made by means of
small electrode points placed at the center of the crystal and kept in
place by a small pressure. An increase in pressure over a moderate
range was found to have no effect on the frequency of the crystal.
The lowering of frequency due to plating was evaluated by depositing
several films of known weight on the crystal and plotting its resonant
frequency as a function of film weight. The intercept of this curve
for a zero plating was taken as the frequency of the unplated crystal.
When the frequency of the oscillator was varied, the current in the
detector showed frequencies of maximum and minimum current out-
put which are respectively the frequencies of resonance and anti-
resonance of the crystal. In order to locate accurately the frequencies
of anti-resonance, it was found necessary to insert a stage of tuning in
the detector, in order to discriminate against the harmonics of the
oscillator. For a given crystal the frequency of the oscillator was
\aried over a wide range and the resonant and anti-resonant frequen-
cies of the crystal were measured. The results of these measurements
are shown by Fig. 25. In this curve the bottom part of the line repre-

432

BELL SYSTEM TECHNICAL JOURNAL

sents the actual measured frequency, while the width of the line is
proportional to the frequency difference between resonance and anti-
resonance. In order to make this quantity observable, the frequency
difference between resonance and anti-resonance is multiplied by a
factor 6.

440

4 320

o 300

Unnncjjj^

*fb

jff

lllllll

Mill

nnniiiiT

■'■'■^■uuxm

EniiW

DDnmiii

"^nofflat

0.2 0.3 0.4 0.5 0.6

RATIO OF OPTICAL TO MECHANICAL

0.7
AXIS

Fig. 25 — Measured resonances of a perpendicularly cut crystal.

As long as the ratio of the optical to the mechanical axis is less than
0.2, the assumption of plane wave motion agrees well with experiment
since there is only one resonance and its frequency does not depend
to any great extent on the optical axis. However, above this point two
frequencies make their appearance and react on each other to produce
the coupled circuit curve shown. Finally when the ratio of optical
to mechanical axes becomes larger a total of four resonant frequenices
appear. Since a large number of crystals are used whose ratios of

ELECTRICAL WAVE FILTERS

433

optical to mechanical axes are greater than 0.2, it becomes a matter of
some importance to investigate the causes of the additional resonances.

Interpretation of the Measured Resonance Frequency Curves of a Per-
pendicularly Cut Crystal
The plane wave assumption is valid for crystals whose width is
less than 1/5 of their length, but it fails for wider crystals. It fails to
represent a rectangular crystal because it does not allow for a wave
motion in any other direction. That such a motion will occur is readily
found by inspecting the stress-strain equations of a quartz crystal,
given by equation (7).

- Xj, = SnX^ + SiiYy + SuZ, + SuY„

- Jy = S12X:, + SuYy + SnZ^ — 5i4Fj,

- z, = Sx^X^ -\- SuYy -\- SzzZ,, (7)

—■ y, = SnXjr — SnYy + -^44^2,

2^ = SuZx -{- SliXy,

Xy = SuZx -T 2(-^ll Sl2)Xy,

where Xx, yy, Zz are the three components of extensional strain, and
Jzy Zx, Xy the three components of shearing strains. X^, Yy, Z^, Y^, Z^,
and Xy are the applied stresses and 5ii, etc. are the six elastic compli-
ances of the crystal. Their values are not determined accurately but
the best known values are given in equation (42). In this equation the
X axis coincides with the electrical axis of the crystal, the Faxis with
the mechanical axis, and the Z axis with the optical axis.

Z AXIS

Y AXIS

Fig. 26 — Form of crystal distorted by an applied Yy force.

Limiting ourselves now to an X or perpendicularly cut crystal the
only stresses applied by the piezo-electric effect are an X^, a Yy, and
a Fj, stress. Hence for such a crystal only four of the six possible types
of motion are excited, three extensional motions Xx, yy, Zz and one shear

434 BELL SYSTEM TECHNICAL JOURNAL

motion y^. Under static conditions, then, the motion at any point in
the crystal is given as the sum of four elementary motions, three ex-
tensional motions and one shear motion. Moreover, these motions are
coupled ^^ as is shown by the fact that a force along one mode produces
displacements in other modes of motion. Figure 26 shows how a
perpendicularly cut crystal will be distorted for an applied Yy force.

Suppose now that an alternating force is applied to the crystal.
The simplest assumption that we can make regarding the motion is
that the motion of any point is composed of four separate plane wave
motions of the four types of vibration and that these react on each
other in the way coupled vibrations are known to act in other mechan-
ical ^^ or electrical circuits. For the present purpose we can neglect
motion along the X or electrical axis since this axis has been assumed
small. The three remaining motions if existing alone will have reso-
nances as shown by the solid lines of Fig. 27. That along the mechan-
ical axis will have a constant frequency, since the mechanical axis is
assumed constant, and is shown by the line C. The extensional motion
along the optical axis will have a frequency inversely proportional to
the length of the optical axis and will be represented by the line A of
the figure. The shear vibration y^, as shown by the section on the
resonance frequency of a crystal vibrating in a shear mode, will have
a frequency varying with dimension as shown by the line B.

In view of the coupling between the motions, the actual measured
frequencies will be as shown by the dotted lines in agreement with well
known coupled theory results.

If we compare these hypothetical curves with the actual measured
values some degree of agreement is apparent. The main resonant
frequency except in the region 0.2 < Ujlm < 0.3 follows the dotted
curve drawn. Also, the extensional motion along the optical axis has a
frequency agreeing with that of Fig. 25. The shear vibration, however,
has an entirely different curve from that conjectured. What is happen-

" The idea of elementary motions in the crystal being coupled together appears
to have been first suggested in a paper by Lack "Observations on Modes of Vibration
and Temperature Coefficients of Quartz Crystal Plates," B. S. T. J., July, 1929,
and was used by him to explain the effect of one mode of motion on the temperature
coefficient of another mode and vice versa. The idea of associating this coupling
with the elastic constants of the crystal occurred to the writer in 1930 but was not
published at that time. It is, however, incorporated in a patent applied for some
time ago on the advantages of crystals cut at certain orientations. More recently
the same idea is given in a paper by E. Giebe and E. Bleckschmidt, Annalen der
Physik, Oct. 16, 1933, Vol. 18, No. 4. They have extended their numerical calcula-
tions to include three modes of motion.

'* This coupling is shown clearly for a mechanical system by one of the few
rigorously solved cases of mechanical motion for two degrees of freedom — the vibra-
tion of a thin cylindrical shell — given by Love in "The Mathematical Theory of
Elasticity," Fourth Edition, page 546.

ELECTRICAL WAVE FILTERS

435

ing there is I think evident from a consideration of Fig. 28. Here in
solid Hnes are drawn two frequency curves one of which, B, is the shear
frequency curve of Fig. 27. The other curve, D, has a rising frequency
with an increase in the optical axis dimension. Assuming these vibra-

>- 320

f?[ 300

220

\\
\ \
\ \

\ \

\
\

\ \
\ \
\ \

■s.

""^-

--^.^

0.2 0.3 0.4 05 0.6 0.7 0.8

RATIO OF OPTICAL TO MECHANICAL AXIS

Fig. 27 — Theoretical resonances of a perpendicularly cut crystal showing effect of

coupling.

tions coupled a resonance frequency curve shown by the dotted line
will be obtained. If this curve is substituted for the shear curve of
Fig. 27 and the actual resonant frequency raised to take account of the
effect of coupling with the longitudinal motion along the mechanical
axis, a curve very similar to the measured curve of Fig. 25 is obtained.
The type of motion coupled to the shear motion is easily found. Its

436

BELL SYSTEM TECHNICAL JOURNAL

480

460

420

400
O

o
o
01 380

360

Z 320

300

280

'"~-v

/ /
//

\ \
\ \
\ \

//
f /

\\
\\

//
1
1
ll

II
1
1
ll

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

RATIO OF OPTICAL TO MECHANICAL AXIS

Fig. 28 — Coupled frequency curve for shear and flexure vibrations.

Fig. 29 — Bar bent in its second flexural mode of vibration.

ELECTRICAL WAVE FILTERS 437

frequency increases as the optical axis dimension is increased and
about the only type of motion which does this is a flexural motion as
shown by Fig. 29. This figure shows the second type of motion possi-
ble to a bar in flexure rather than the first for experiments by Harrison ^^
show that the frequency for the first type of motion is too low to ac-
count for this vibration./ Harrison has also measured the frequencies
of a bar in its second flexural mode and the solid line, D, of Fig. 28 is an
actual plot of these measured frequencies up to a ratio of hllm = 0.25,
which is as far as Harrison carries his measurements. The rest of the
curve is obtained by extrapolation. There is no doubt then that a
flexural motion is involved in this coupling. The mechanism by which
the bar is driven in flexure will be evident if we observe what happens
to a square on the crystal in the unstrained state. As shown by Fig.
29, its deformation is similar to that of a shear deformation. The
amount of shear depends on the distance from the nodes of the crystal.
Some of the shear is in one direction and some in the other but the
two amounts are not balanced and hence a pure shear in one direction
can excite a flexural motion of the crystal.

The strength of the coupling from the mechanical axis motion jy to
the shear motion y^, and the extensional motion along the optical axis
Zz are indicated by the coupling compliances SuHs'^iSa and SizHsi^Sss,
respectively. From the values of these constants we find that the
shear motion is more closely coupled than the z extensional motion,
and this is indicated experimentally by the greater width of the shear
line.

Effect of a Rotation of the Longest Axis with Respect to the Electrical

Axis on the Resonances of a Crystal

From the qualitative explanation of the secondary resonances

given above, it is possible to predict how these resonances will be

affected by any change in the crystal which changes the constants

determining the three modes of motion and their coupling coefficients.

One method for varying these constants is to change the direction for

cutting the crystal slab from the natural crystal. In the present paper

consideration is limited to those crystals which have their major faces

perpendicular to an electrical axis, i.e., a perpendicularly cut crystal

with its longest direction rotated by an angle 6 from the direction of the

mechanical axis. The convention is adopted that a positive angle is a

clockwise rotation of the principal axis for a right handed crystal,

when the electrically positive face (determined by a squeeze) is up.

15 " Piezo-Electric Resonance and Oscillatory Phenomena with Flexural Vibration
in Quartz Plates," J. R. Harrison, /. R. E., December, 1927.

438

BELL SYSTEM TECHNICAL JOURNAL

For a left-handed crystal a positive angle is in a counter-clockwise
direction.

In the section dealing with elastic and piezo-electric constants for
rotated crystals (page 449) is given a method for determining the elastic
constants of a rotated crystal and curves are given for the ten elastic
constants. These have been worked out by Mr. R. A. Sykes of the
Laboratories. The method of designation is the following: The X
axis remains fixed and is designated by 1'. The axis of greatest length
is designated by 2' , since in the unrotated crystal the mechanical axis,
corresponding to the F direction, is the axis of greatest length. Exten-
sional motion perpendicular to the 2' axis is designanted by 3', and
shear motion in the plane determined by the 2', 3' axes is designated by
4'. The ten resulting constants ^n', 522', 533', 544', 512', ^13', Sm' s^',
s<ii , Szi are shown evaluated in terms of the angle d on Fig. 30. Since

5 10

s'44

■^

S'33

N ■

N,,

-^11

— \

^--

■s^

■

— "

^-v

■V

/■

' —

S|4

-~>

-/-

c — -p

^lN

^'34

■^

<tf*

1 — 1

=-::

ii24

[siT

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90
ANGLE OF ROTATION IN DEGREES

Fig. 30 — Elastic compliances of a perpendicularly cut crystal as a function of the

angle of rotation.

motion and coupling to motion along the X axis can be neglected, the
constants of interest are 522', S33', Sa', 523', ^24', ^34'. Since the 2' or 3;'

ELECTRICAL WAVE FILTERS

439

380
370

\\
\\
\»
\\
\\

360

\
\

q; 330
LLl

1-
UJ

S^ Tin

_l
o

-I

LLl
IL

..11. J

....1

250
240

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

RATIO I

Fig. 31^Measured resonances oi a d = — 18.5° perpendicularly cut crystal.

440

BELL SYSTEM TECHNICAL JOURNAL

axis is the principal axis of motion, the mutual compliances of principal
interest are 523', determining the coupling between the Y' extensional
motion and the Z' extensional motion, and 524' determining the coupling
between the Y' extensional motion and the Y^' shear motion. It is
the shear motion which is most objectionable, because it is more highly-
coupled than the Z' extensional motion, because it is lower in frequency,
and because it is coupled to a flexural mode. Hence, if this motion
can be eliminated or made very small, a much better crystal for most
purposes is obtained. We note that if d is —18.5° or if 0 = 41.5° the
shear coupling coefficient vanishes and hence a force in the Yy direction
produces no y^ shear or vice versa. Of these the —18.5° crystal is
driven more strongly by the piezo-electric effect and hence has a more
prominent resonance.

390

a. 330

290

230

\
\

\ ^

/ D

B ^

^^^as

"rm^

0.1

0.2

0.3

0.4

0.5 0.6

0.7

0.8

0.9

1.0

• RATIO filO)

Fig. 32 — Measured resonances of a 0 = + 18.5° perpendicularly cut crystal.

Accordingly the resonances of a 0 = —18.5° cut crystal have been
measured in a similar way to the 0 = 0° cut crystal shown in Fig. 25.
The result is shown on Fig. 31. As will be seen from the figure, the

ELECTRICAL WAVE FILTERS

441

shear resonance indicated by B is barely noticeable, while the z exten-
sional mode indicated by A is somewhat stronger although higher in
frequency. The frequency of the principal mode is not greatly affected
by an increase in the z' axis until the ratio of axes is greater than .6.
Another angle of some interest is 0 = + 18.5° since there the z' ex-
tensional coupling disappears. The resulting resonances are shown on
Fig. 32. It will be noted that the z' extensional resonance curve A is
very weak, while the shear curve B is quite pronounced.

A n Equivalent Electrical Circuit for a Crystal Possessing Two Degrees

of Motion

The above explanation accounts qualitatively for all the resonances
observed in the crystal and how they are varied by a rotation of the
crystal. It is desirable, however, to see if a quantitative check can be
obtained from the known elastic constants of the crystal. To obtain
a complete check would require a system capable of five degrees of
motion. However, if we take the simplest case, the — 18.5 degree cut
crystal, only two modes of motion have to be considered, and even for
the zero cut crystal, a good agreement is obtained by lumping the
shear and extensional mode as one mode of motion and considering its
reaction on the fundamental mode. Hence consideration is limited
in this paper to a circuit having two modes of motion.

The properties of a single mode of motion can be represented for
frequencies which do not exceed the first resonant frequency of the
crystal, by the simple electrical circuit of Fig. ?)?)A. Here the capaci-

I — "w^ — ^H

""Y

I — ^W^

-Cm Cz

i-z

■^M^ — I

-Cm

A B

Fig. 2)i — Equivalent electrical circuit of a crystal having two modes of motion.

tance represents the mechanical compliance of the bar, the charge on the
condenser represents a displacement per unit length of the bar, while
the current flowing through the circuit represents the velocity of a
point on the bar. The inductance represents the mass reaction of the
crystal. The representation of the motion of a bar by a simple lumped
circuit assumes that the bar moves as a whole, that is, if a force is
applied to the body it contracts or expands equally at all parts of the

442 BELL SYSTEM TECHNICAL JOURNAL

bar. This is contrary to actual conditions, since expansions or con-
tractions proceed in the form of a wave from the ends of the bar toward
the center. However, if consideration is Hmited to low frequencies,
i.e. frequencies which do not exceed by much the first resonance of the
bar, the approximation is good and a considerable simplification in the
analysis is made. To take account of wave motion, the representation
has to be an electric line as was pointed out in connection with acoustic
filters.16

To represent two separate modes of motion and their coupling, the
circuit shown by Fig. 2>d)B is employed. A little consideration shows
that the type of coupling existing in a crystal is capacitative since an
extension along the mechanical axis produces a contraction along the
optical axis, and vice versa. Since strains in mechanical terms are
equivalent to charges in electrical terms, this type of coupling can be
represented only by a capacitative network. This representation is
entirely analogous to the T network representation for a transformer.^^
The constants of the network can be evaluated in terms of the elastic
constants of the crystal as follows: For a —18.5 degree cut crystal,
we can write the stress strain equation (7) as

Jy = 522'F, + 523%, (8)

since we are neglecting motion along the X axis and since ^24', the
coupling coefficient of the shear to the Y' axis is zero. No Y^ force is
assumed acting. If we work out the equation for the charges on the
condensers of the equivalent representation shown in Fig. ZZB we have,
with the charges and voltages directed as shown

Qi = 1 ^. + e.

1 - K^ ' M - X^ '

, (9)

„ ^ ey\CyC,K e^Cz
^' 1 - i^2 -t- ^ _ ^2 ,

where K the coupling factor between the two modes of motion, is de-
fined by the relation

K = ^^ . (10)

Associating Qi with jy, the displacement per unit length, Q^ with

1^ See "Regular Combination of Acoustic Elements," W. P. Mason, B. S. T. /.,
April, 1927, p. 258.

1^ See, for example, p. 281 in the book "Transmission Circuits for Telephonic
Communication" by K. S. Johnson.

ELECTRICAL WAVE FILTERS

443

z^, By with Yy and e^ with Z., we have on comparing (9) with (8)

522

1 - K^

; ^23 —

1 - K^

; ^33

1 - K^

or inversely

Cj/ — 522

1 -

523

522 533

533

523

522 533

\K =

523

A 522 533

(11)

(12)

If, now, alternating forces are appUed to the crystal, another reac-
tion to the applied force enters, namely the mass reaction of the crystal
due to the inertia of the different parts. To take account of this reac-
tion, the inductances are added to the two meshes representing mass
reaction for the two modes of motion. To determine the value of the
inductance, consider first the representation for one mode of motion
shown by Fig. i^A. The resonant frequency of the system is given by

Itt^LC

On the other hand, the resonant frequency of a bar is given by

f^_ 1_,

(13)

(14)

where / is the length of the bar, 5 its compliance, and p its density.
But in the above representation the capacitance C is the compliance
constant 5 so that, on comparing (13) and (14) we find

TT"

(15)

In a similar manner for the coupled circuit. Fig. 33B, there results

T -Hp.T -

(16)

where ly is the length of the crystal in centimeters along the y axis, and
Iz the length of the crystal in centimeters along the z axis. Hence all
of the constants of Fig. 33B, which represents the crystal for mechanical
vibrations subject to the restrictions noted above, are determined and
we should be able to predict all of the quantities which depend only
on the mechanical constants of the crystal.

Of these the most important are the resonance frequencies of the
crystal and their dependence on dimension, temperature coefficient and
the like. To determine the natural mechanical resonance of a crystal,

444 BELL SYSTEM TECHNICAL JOURNAL

we solve the network of Fig. ^2)B to find the frequencies of zero im-
pedance for either an appUed Yy force or an applied Z-, force. The
result is two frequencies /i and/2 given by the coupled circuit equations

(17)

where

IwyLyCy liryLzCg

Then /a and Jb represent the natural frequencies along the Y and Z
axis respectively when these two motions are not coupled together.

Two limiting cases of interest are obtainable from these relations.
If /b is much larger than /a, the equations reduce to

/i =/aV1 -K' =

f^ = fj^ = ^

2k^pszz'[_\ — S'iz'^-jsii.'szz'']

upon substituting the value of the constants given before. The first
equation shows that for a long thin rod the frequency depends on
the elastic constant 522', which is the inverse of Young's modulus.
For the frequency J2, which corresponds to that of a thin plate, a
different elastic constant appears. Upon evaluating the expression
■^33'[1 — S2z''^js22'szz'^ in terms of the elastic constants which express
the forces in terms of the strains — see equation (25) — we find that
5i3'(l — Siz'^lsii'szz') = I/C33. C33 measures the ratio of force to strain
when all the other coupling coefficients are set equal to zero, and
corresponds to the frequency of one mode vibrating by itself without
coupling to other modes. Hence the frequency of a thin plate should
be

f-^T' ^'")

where c„„ represents the elastic coefficient for the mode of motion con-
sidered, and t is the thickness of the plate. This deduction has been
verified by experimental tests on thin plates.

Let us consider now the curves for the —18.5 degree cut crystal
shown by Fig. 31. The values of the elastic constants for this case are

522' = 144 X 10-1^ cm.Vdynes; 523' = - 21.0 X IQ-i^

533' = 92.5 X 10-1^ (21)

ELECTRICAL WAVE FILTERS

445

Hence from equations (17), (18) and (21) one should be able to check
the measured frequency curves of Fig. 31. The result is shown on the
dotted lines of these curves. The agreement is quite good although a
slightly better agreement would be obtained if 523 had a smaller value.
Since these constants have never been measured with great accuracy,
it is possible that they deviate somewhat from the curves of Fig. 30.

This theory can be applied also to a 0 = + 18.5 degree cut crystal
since the extensional coupling coefficient vanishes for this angle. The
agreement is quite good if the frequency for the uncoupled mode given
by the section on vibration in shear mode (page 446) is used in place of
equation (14). The resonances for the 0 = 0° cut crystal shown by
Fig. 25 cannot be accounted for quantitatively by the simple theory
given here since there are three modes of motion operating. The
shear mode of motion is more closely coupled to the principal mode
than is the Z^ extensional mode and hence a fair approximation is ob-
tained by considering only the shear mode. However, for complete
agreement the theory should be extended to a triply coupled circuit and
that is not done in this paper.

Another phenomenon of interest which can be accounted for by the
circuit of Fig. 335 is the temperature coefficient of the crystal and its
variation with different ratios of axes and different angles of rotation.
To obtain the relation, we assume that each of the vibrations may have
a temperature coefficient of its own as may also the coefficient of
coupling K. If a small change of temperature occurs, /i will change
to /i(l + T^T), /a to /^(l + TaAT), Jb to /^(l + TbAT) and K to
K{1 + Tk^T). Assuming AT" small so that its squares and higher
powers can be neglected, we find from equation (17) that

r = ^

TaJa'

1 +

fsW -2K') -/a'

//(I + 2K'

+ Tb/b'

Ia'

<Ub'

fA^y + ^KjAjs'^
2TKfAjB'

^{fB'-fA'y-i-^KjAjB'}

(22)

The temperature coefficients of the six elastic constants have been
measured at the Laboratories ^^ by measuring the frequency tempera-
ture coefficients of variously oriented crystals. The temperature
coefficients of the six elastic constants can be calculated from these

'* These coefficients have been evaluated in cooperation with Messrs. F. R.
Lack, G. W. Willard and I. E. Fair and their work is discussed in detail in their
compani6n paper in this issue of the B. S. T. J.

446 BELL SYSTEM TECHNICAL JOURNAL

measurements and have been found to be, in parts per million per
degree centigrade:

r.^^ = + 13; r,., = - 1230; T.^^ = - 347; (23)

r.^, = + 130; r.33 = + 213; T,^ = + 172.

Using these values and neglecting the extensional motion Z^, the tem-
perature coefficients calculated from equation (22) for a 0 degree cut
crystal are shown on the dotted line of Fig. 5 and agree quite well with
the measured values.

The Resonance Frequencies of a Crystal Vibrating in a Shear Mode
The equations of motion for any aelotropic body are
d^u ^ dX^ dXj dX.

df dx By dz

BY.

_j_

BYy

BY.

^~dt'~

Bz •

B'^w
'Bt^ ~

BZ.
Bx

BZy

BZ,

Bz •

(24)

where u, v, w are the displacements of any point in the crystal along the
X, y, z axes respectively and X., etc. are the six applied stresses. The
strains have been expressed in terms of the stresses by equation (7).
It is more advantageous for the present purpose to express the stresses
in terms of the strains, which can be done by the following equations:

Xx = CnXx + Cuyy + Cisz^ + Cuyz,

Yy = Ci2X^ + C22yy + CisZ^ + C24>'2,

Z^ = CisXx + Cosyy + CssZ^, (25)

Yz = CuXx -\- C24yy + Cuyz,

z^ X ^ c^iZx ~r CiiXy,

Xy = CliZx + 2(^11 ~ Cl2)Xy,

where the c's are the elastic constants and the strains Xx, etc., are given
in terms of the displacements u, v, w by the equations

Bu Bv Bw I Bv , Bw

Bx'-"" By' ' Bz ' ^' \Bz^ By

-^ + -^j;x, - y^ + ^j-

(26)

In equation (24) there exist the reciprocal relations

Xy = Yx; X, = Zx\ F, = Zy. ill)

ELECTRICAL WAVE FILTERS 447

For a free edge, i.e. no resulting forces being applied to the crystal, the
conditions existing for every point of the boundaries are

Xt, = Xx cos {v,x) + Xy cos {v,y) + X^ cos {v,z) = 0,
Y, = Yx cos {v,x) + Yy cos {v,y) + Y^ cos (j/.z) = 0, (28)
Zy = Zx cos {v,x) + Zy cos {y,y) -\- Z^ cos {y^z) = 0,

where j' is the normal to the boundary under consideration.

If these equations are combined and completely solved, the motion
of a quartz crystal is completely determined. The results which were
obtained above in an approximate manner could be rigorously solved.
However, on account of the difficulty ^^ of the solution, this is not at-
tempted here. In the present section it is simply the purpose to find
out what resonances a crystal will have if it is vibrating in a shear mode
only. To avoid setting up motion in the other modes of vibration, the
coupling elasticities Cu, C24, C34 are assumed zero. Similarly if C\i,
Ci3, C23 were set equal to zero we should have the possibility of three
extensional modes and one shear mode vibrating simultaneously with
no reaction on one another, and the equation of motion would be

P -^ = ^ (C22yy) + J Lcay,'], (29)

The displacements u, v, and w would be the sum of the displacements

caused by the four motions. To find the displacements and resonances

caused by the shear mode yz, we neglect the other modes and have the

equations

d^v ^ d , .

"^^ ^^ (30)

d^W d , .

'-^^"^ '''Vy^^'^'

•J

Differentiating the first of equations (30) by — , and the second by

-r- , and adding, there results,
dy

d"^ / dv dw\ _

dy- dz^

(31)

1' For example if motion is limited to the y and s directions, and the coefficient Cu
is set equal to zero, the equations reduce approximately to those for a plate bent in
flexure, and this case has never been solved for the boundary condition of interest
here, namelv all four edges being free to move — see Rayleigh "Theory of Sound,"
page 372, Vol. I, 1923 edition.

448

BELL SYSTEM TECHNICAL JOURNAL

Since— + -7- = jz, this reduces to
dz dy

(32)

where c^ = CuIp-

For a simple harmonic vibration, of frequency /, the equation
reduces to

2 "I" 110.2

_ dy

dz^

{S3)

where p = It/. The solution of this equation consistent with the
boundary conditions (28) is

ZZA,

. miry . nwz

sm sm — J—

a 0

cos pt,

(34)

where a is the length of the crystal in the y direction, h the length of the
crystal in the z direction, and m and n are integers. Substituting this
equation in the equation (32), we find that it is a solution provided

(2^fy

Hence the resonant frequencies of the crystal in shear vibration are

(35)

(36)

To find the shape of the deformed crystal, we have from (30) for
simple harmonic vibrationNl,

V = —

W = rs

p^ dz
c2 dyz

- a'^P

dyz

m'^TT^b'^ + n^TT^a^ dz

- a'b^ dy.

(37)

p^ dy m^TrW + n^TT^a^ dy

The cases m = 0, n = 1 and m = 1, n = 0 require a stress known
as a simple shear to excite them, whereas the stress applied by the
piezo-electric effect is a pure shear. Hence the case m = I, n = 1
provides the lowest frequency solution. The displacements v and w
for this case are by equations (34), (37) and (38)

— a^bir I . -wy TTZ
^ ~ 9 9 I — ^r^ sm -^ cos -7-

ab^TT / Try . TTZ

"^ — T~r~i — 513 cos -^ sm -7-

ir^a^ + T^ b~ \ a b

(39)

The resulting distortion of the crystal is shown by Fig. 34.

ELECTRICAL WAVE FILTERS

449

We can conclude, therefore, that the solution for a shear vibration
in a quartz crystal will be given by equation (34). It is obvious from
Fig. 34 that the shear vibration will have a strong coupling to a bar

1
I

\
\
\

-'"'

\
\
\
\

\
\

\
1

1 i

1 1

Fig. 34— Form of crystal in shear vibration.

bent in its second mode of flexure, since the form of the bent bar, as
shown by Fig. 29, is very closely the same as a given displacement line
in the crystal vibrating in shear. Little coupling should exist between
the shear mode and a bar in its first flexure mode, since this mode of
flexure requires a displacement which is symmetrical on both sides of
the central line whereas the bar vibrating in shear has a motion in
which the displacement on one side of the center line is the opposite
of the displacement on the other side of the center line.

The Elastic and Piezo-Electric Constants of Quartz for Rotated Crystals -^
W. Voigt ^^ gives for the stress strain and piezo-electric relation in a
quartz crystal, for the three extensions and one shear found above,

— Xx = SxiXx + SiiYy + SxzZi + SuY^,

— jy = siiXx + 522^2, + snZz + SiiY:,

— Zz = SizXx + SizYy + SzzZz 4" SziYz,

— Jz = ^14-^1 + S2iYy + SziZz + Si^Yz,

— Px = diiXx + diiYy + dnZz + dxiYz,

(40)

(41)

2" The material of this section was first derived by Mr. R. A. Sykes of the Bell
Telephone Laboratories.

^' W. V'cigt, Lehrbach Der Kristallphysik.

450 BELL SYSTEM TECHNICAL JOURNAL

where

Xx, Jy, z, = extensional strains = elongation per unit length,
Xx, Yy, Zz = extensional stresses = force per unit area,
yz = shearing strain = cos of an angle,
Y, = shearing stress = force per unit area,
Sij = elastic compliances = displacement per dyne,
dij = piezo-electric constants = e.s.u. charge per dyne,
Px = piezo-electric polarization = charge per unit area.

The best measured values for these constants when the X axis coin-
cides with the electric axis of the crystal, the Faxis with the mechanical
axis and the Z axis with the optical axis, are

^11 = ^22 = 127.2 X 10-" cm.Vdyne,

5i2 = - 16.6 X 10-14 cm.Vdyne,

sn = S23 = - 15.2 X 10-1" cm.Vdyne,

^24 = - su = 43.1 X 10-14 cm.Vdyne,

^33 = 97.2 X 10-14 cm.Vdyne,

^34 = 0, (42)

544 = 200.5 X 10-14 cm.Vdyne,

^ ^ /c c!A v/ m-R^-S-u. charge

an = — di2 = — 6.36 X 10^ -. 2_ , .

dyne

^13 = 0

^:4 = 1.69 X 10-B "•^•": "^^'"g^^

dyne

If, now, we maintain the direction of the electrical axis but rotate
the direction of the principal axis by some angle d, the resulting con-
stants of equations (40) and (41) undergo a change.

Let the direction cosines for the new axes be given by

(43)

The convention is adopted that a positive angle 0 is a clockwise rota-
tion of the principal axis of the crystal, when the electrically positive
face (determined by a squeeze) is up. For a left-handed crystal a
positive angle is in a counter clockwise direction, d is the angle be-
tween the previously unprimed and the primed axes.

«i

nii

712

niz

ELECTRICAL WAVE FILTERS

451

If we transform only the y and z axes, there results
h = h = nil = ni = 0,
h = 1,
fUi = fis = cos d,
— 712 = ni3 = sin 9.
Love ^^ gives the transformation for the stress and the strain func-

(44)

tions as

= xj,^ + jym^ + z^n^ + y^mxn\,

= Xxl^ + yymi + 22^2^ + yzm^Ui,

= Xxh^ + y^Ws^ + z.nz^ + yzmsfis,

= 2:j£;J2/3 + 2yym2m3 + 2z^n2nz + yzim^nz + m3«2),

= Xx/i^ + Fj^wr + ^zWi^ + F22wiWi,

= Xx^2^ + Fym2- + Z^7Z2" + F22W2W2,

= Xxh-" + F,W32 + Z,«3- + F,2m3W3,

= Xxhh + Yytn^nis + Z^n^Uz + Y^intons + m3W2).

(45)

(46)

Substituting (44) in (45) and (46) and then expressing Xx, yy - • •
Xx, Yy . . ., etc., in terms of Xx\ yy' . . . Xx', Yy' . . ., etc., we
may substitute these values in equations (40) and (41) to give the stress-
strain and polarization in a crystal whose rectangular axes do not coin-
cide with the real optical and mechanical axis. Performing the above
operations, a new set of constants Si/, are obtained which are functions
of 6, namely:

•^11 = Su,

■^12' = |[5i2 + ^13 + (-^12 - ^13) cos 26 — Sii sin 20],

•^13' = hL^iz + S12 + (5i3 - ^12) cos 26 + su sin 26~],

Sn' = Sii cos 26 + (512 — ^13) sin 26,

S22' = ^11 cos^ 6 -f 533 sin^ 6 -f 7su cos^ 6 sin 6

+ (25i3 + Sii) sin^ 6 cos^ 6,

S23' = 5i3(cos'' 6 + sin" 6) + Su{s'm^ 6 cos 6 - cos^ 6 sin 6)

+ {su + .^33 — 544) sin^ 6 cos^ 6,
S2i' = — 5i4(cos" 9 — 3 sin^ 6 cos^ 6) + (25ii — 2^13 — Sa) cos^ 6 sin 0

+ (2^13 — 2^33 + Sii) sin^ 9 cos 9,

533' = -^33 cos'' 6 + 5ii sin"* 9 — 2sii sin^ 9 cos 0

-f- (25i3 + -^44) sin- 6 cos^ 6,

22 "The Mathematical Theory of Elasticity," Cambridge University Press, pp.
42 and 78.

452

BELL SYSTEM TECHNICAL JOURNAL

Ssi = 5i4(sin^ 6 — 2> sin^ 6 cos^ 6) + (2^11 — 2sn — ^44) sin^ d cos d

+ (2^13 — 2^33 + 544) cos^ 6 sin d,
544' = (4533 + 45ii — 8^13 — 2^44) sin^ 6 cos^ 6 + 4^14

X (sin^ 0 cos ^ - sin 6 cos^ d) + 544(sin'' 6 + cos" d)

and

dii = dn,

du = - i[^u(l + cos 26) + (/i4 sin 2^],

f^is' = - |[c?ii(l - cos 2d) - dii sin 2dJ,

dii = ^14 cos 26 — dn sin 26.

The curves representing the 5' values for varying angles of orienta-
tion are plotted on Fig. 30 while the values of d' are plotted on Fig. 35.

^-'^

^=^13

■^

^-d' y

7--d,4/

-t-"

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40

ANGLE OF ROTATION IN DEGREES

60 70 80 90

Fig. 35 — Piezo-electric constants of a perpendicularly cut crystal as a function of the

angle of rotation.

Some Improvements in Quartz Crystal Circuit Elements

By F. R. LACK, G. W. WILLARD, and I. E. FAIR

The characteristics of the F-cut quartz crystal plate are discussed.
It is shown that by rotating a plate about the X axis special orientations
are found for which the frequency spectrum is simplified, the temperature
coefficient of frequency is reduced practically to zero and the amount of
power that can be controlled without fracture of the crystal is increased.
These improvements are obtained without sacrificing the advantages of the
Y cut plate, i.e., activity and the possibility of rigid clamping in the holder.

THERE are at the present time two types of crystal quartz plates
in general use as circuit elements for frequency stabilization at
radio frequencies, namely, the X-cut and F-cut. ^ This paper is
concerned with the improved characteristics of plates having radically
new orientations.

In its usual form the F-cut plate is cut from the mother crystal,
as shown in Fig. 1. The electric field is applied along the F direction

OPTIC AXIS

MECHANICAL
AXIS

Fig. 1 — Showing relation of Y-cut quartz crystal to the crystallographic axes.

and for high frequencies an Xy shear vibration is utilized.

The frequency of such a vibration is given by the expression:

1 " Piezo-Electric Terminology," W. G. Cady, Proc. I. R. E., 1930, p. 2136.

453

454 BELL SYSTEM TECHNICAL JOURNAL

where Cee = the elastic constant for quartz connecting the Xy stress
with an Xy strain = 39.1 X 10^" dynes per cm.^
P = the density of quartz = 2.65 gms. per cm.^
/ = the thickness in cm.

On substituting the numerical values in equation (1), a frequency-
thickness constant of 192 kc. per cm. is obtained which checks within
3 per cent the value of this constant found by experiment.

This Xy shear vibration is not appreciably affected when the
plate is rigidly clamped, the clamping being applied either around the
periphery if the plate is circular, or at the corners if square. Hence a
mechanically rigid holder arrangement is possible which is particularly
suitable for mobile radio applications.^

The temperature coefficient of frequency of this vibration is approxi-
mately + 85 parts/million/C.°, which means that for most applications
it must be used in a thermostatically controlled oven. In operation,
this comparatively large temperature coefficient is responsible for a
major part of any frequency deviations from the assigned value.

Another important characteristic of the F-cut crystal is the secon-
dary frequency spectrum of the plate. This secondary spectrum con-
sists of overtones of low frequency vibrations which are mechanically
coupled to the desired vibration and cause discontinuities in the
characteristic frequency-temperature and frequency-thickness curves
of the crystal. In some instances these coupled secondary vibrations
can be utilized to produce a low temperature coefffcient over a limited
temperature range.' But in general, at the higher frequencies (above
one megacycle) this secondary spectrum is a source of considerable
annoyance, not only in the initial preparation of a plate for a given
frequency but in the added necessity for some form of temperature
control. In practice, these plates are so adjusted that there are no
discontinuities in the frequency-temperature characteristic in the
region where they are expected to operate, but at high frequencies it
is difficult to eliminate all of these discontinuities over a wide temper-
ature range. If, then, for any reason the crystal must be operated
without the temperature control, a frequency discontinuity with
temperature may cause a large frequency shift greatly in excess of
that to be expected from the normal temperature coefficient.

From the above considerations it may be concluded that the
standard F-cut plate has two distinct disadvantages: namely, a

2U. S. Patent No. 1883111, G. M. Thurston, Oct. 18, 1932. "Application of
Quartz Plates to Radio Transmitters," O. M. Hovgaard, Proc. I. R. E., 1932, p. 767.

3 "Observations on Modes of Vibrations and Temperature Coefficients of Quartz
Plates," F. R. Lack, Proc. L R. E., 1929, p. 1123; Bell Sys. Tech. Jour., July, 1929.

QUARTZ CRYSTAL CIRCUIT ELEMENTS 455

temperature coefficient requiring close temperature regulation and a
troublesome secondary frequency spectrum. Assuming that the
temperature coefficient of the desired frequency could be materially
reduced, the effect of any secondary spectrum must also be minimized
before temperature regulation can be abandoned. In fact, from the
standpoint of satisfactory production and operation of these crystal
plates, it is perhaps more important that the secondary spectrum be
eliminated than that the temperature coefficient be reduced.

The Secondary Spectrum

The secondary spectrum of these plates, as has been indicated
above, is caused by vibrations of the same or of other types than the
wanted vibration taking place in other directions of the plate and
coupled to the wanted vibration mechanically. This condition of
affairs exists in all mechanical vibrating systems but is complicated
in the case of quartz by the complex nature of the elastic system
involved.

Considering specifically the case of the F-cut plate the desired
vibration is set up through the medium of an Xy strain. Hence any
coupled secondary vibrations must be set in motion through this Xy
strain. Referring to the following elastic equations for quartz (in
these equations X, Y and Z are directions coincident with the crystallo-
graphic axes; see Fig. 1 and Appendix),

Xx = CuXs + Cnyy + CnZ^ + Cuy^

Yy = Cl2Xj; + Ciljy + C^^Zz + C^iJ z

Zz = CxzXx + Ci^Jy + CzzZz

Yz = CnXx + Ciiyy + cajz

Zx = ~r Css^^x -\- c^^Xy

Xy = + CseZx ~t" C^^Xy

(2)

it will be seen that by reason of the constant C56 an Xy strain will set
up a stress in the Zx plane which in turn will produce a Zx strain.
Hence the .y^ and z^ strains are coupled together mechanically, the
value of the constant c^& being a measure of that coupling.

High order overtones of vibrations resulting from this Zx strain
constitute the major part of the secondary frequency spectrum of
these plates.

The technique for dealing with this secondary spectrum in the past
has been the proper choice of dimensions. At high frequencies these
overtones occur very close together and when one set is moved out
of the range by grinding a given dimension another set will appear.
Some benefit is obtained with the clamped holder, which tends to
inhibit certain types of transverse vibrations; but as indicated above.

456

BELL SYSTEM TECHNICAL JOURNAL

12
10
6
6
4

2
0

XIO-®

20
15
10
5

<o
-•o -5
u

-10
-15

xioio

BC-CUT

Y-CUT

AC-CUT^

xioio

— NOTE —
^16=^26 = ^36 = 046=0

FOR ALL VALUES

OF e

MfY-

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90

ANGLE OF ROTATION ABOUT X AXIS IN DEGREES (G)

Fig. 2 — coe', Cm, and dae' as a function of rotation about the X axis.

QUARTZ CRYSTAL CIRCUIT ELEMENTS 457

the elimination of the effects of the coupled secondary frequency
spectrum over a wide temperature range is a difhcult matter.

Another method has been developed recently for dealing with these
coupled vibrations."* This consists of reducing, by a change in
orientation, the magnitude of the elastic constant responsible for the
coupling. If the orientation of the crystal plate be shifted with
respect to the crystallographic axes then, in general, the elastic con-
stants with reference to the axes of the plate will vary. The direct
constants (ci/, C22', •••) which represent the longitudinal and shear
moduli will of course vary in magnitude only, while the cross constants
(ct/, • • •) will vary both as to magnitude and sign. There is a possi-
bility therefore that the proper choice of orientation of the plate will
reduce Ci,e to zero without at the same time introducing other couplings.

Figure 2 shows graphically the variation of c^e and C5& as a function
of rotation about the X axis. These have been calculated by means
of the equations given in the appendix. It will be seen that at ap-
proximately + 31° and — 60° rse' becomes zero. Here then are two
orientations for which the coupling between the Xy and Zx strains
should be zero.

In shifting the orientation of the plate the necessity for exciting the
wanted vibration piezo-electrically must not be lost sight of. Hence
in addition to computing the values of the elastic constants the
variation of the piezo-electric moduli as a function of orientation must
also be examined. Figure 2 also shows the effect of rotation about
the X axis on d26 (the constant connecting the Ey' electric field with
the Xy strain). It will be seen that at both + 31° and — 60° the Xy
vibration can be excited piezo-electrically but it is to be expected that
a plate cut at — 60° will be relatively inactive,^ for di^' at this point
is only 20 per cent of its value for the F-cut plate. On the other
hand at -f 31° a plate would be practically equivalent to the F-cut
as far as activity is concerned. The frequency of the Xy vibration
for these special orientations can be calculated by means of equation
(1) substituting for Cee the value of Cee' for the given angle as read
from the curve of Fig. 2.

* The expression of the coupling between two modes of vibration in quartz in
terms of the elastic constants was first suggested in 1930 by Mr. W. P. Mason of
the Bell Telephone Laboratories.

* The word "activity" is a rather loose term used by experimenters in this field
to describe the ease with which a given vibration can be excited in a particular circuit.
It is often spoken of in terms of the grid current that is obtained in that circuit
or the amount of feedback necessary to produce oscillation. It can better be ex-
pressed quantitatively as the coupling between the electrical and mechanical systems
(not to be confused with the mechanical coupling between different vibrations
described above) which is a simple function of the piezo-electric and elastic moduli
of the vibration involved and the dielectric constant of the crystal plate.

458

BELL SYSTEM TECHNICAL JOURNAL

For the purpose of identification the plate cut at +31° has been
designated as the ^C-cut and the plate cut at — 60° the BC-cut.
Crystal plates having these orientations have been made up and
tested. It is evident from the frequency-temperature and frequency-
thickness characteristics of both cuts that a simplification of the
frequency spectrum results from the reduction in coupling to secondary

9000
8500
8000
7500
7000

Q 6500

z
o

JJj 6000

LlJ

Q. 5500

01
UJ

d 5000

>-
u

1 4500

5 4O00

V 3500
O

2
UJ

D 3000
Cf

"- 2500

2000

1500

1000

500

j^-

"^^

Y-CUT/

/''

"aC-CUT (31°)

■"^

20 24 28

32 36 40 44 48 52 56 60 64
TEMPERATURE IN DEGREES CENTIGRADE

72 76

P'ig. 3 — ^Frequency-temperature characteristics of AC-cut and Y-cut plates of same

frequency and area.
Frequency 1600 KC.
Dimensions:

Y-cut y =1.22 mm. x = 38 mm. c = 38 mm.

AC-cut (31°) y' = 1.00 mm. x = 38 mm. z' = 38 mm.

QUARTZ CRYSTAL CIRCUIT ELEMENTS 459

vibrations. Frequency discontinuities of the order of a kilocycle or
more which are a common occurrence with the F-cut plate have
disappeared and frequency-temperature curves that are linear over a
considerable temperature range can be obtained without much diffi-
culty. This is illustrated by Fig. 3 which shows frequency-tempera-
ture characteristics for both ^C-cut and F-cut plates of the same
frequency and area. The ^C-cut plate can be clamped to the same
extent as the F-cut plate.

. There is still some coupling remaining to certain secondary fre-
quencies. These frequencies are difficult to identify but are thought
to be caused by overtones of fiexural vibrations set up by the x/
shear itself and hence would be unaffected by the reduction of c^/.
These remaining frequencies do not cause much difficulty above 500 kc.
For the ^C-cut (-f 31°) plate the temperature coefficient of frequency
is + 20 cycles/million /C.°, while for the -BC-cut (— 60°) plate it is
— 20 cycles/million/C.°.

In addition to these calculations for the Xy' vibration in plates
rotated about the X axis, a detailed study has been made of other
types of vibration and rotation about the other axes. For high
frequencies nothing has been found to compare with the reduction in
complexity of frequency spectrum obtainable with these two orienta-
tions.

Temperature Coefficients

This study has produced in the AC-cut a new type of plate
which has superior characteristics to the standard F-cut: i.e., a simpli-
fied frequency spectrum and a lower temperature coefficient. The
values of the temperature coefficients obtained for these new orienta-
tions are significant and suggest that perhaps other orientations can
be chosen for which the temperature coefficient will be zero. With
the measured values of the temperature coefficients for the different
orientations and the Cee equation (Appendix) it is possible to compute
the temperature coefficient for any angle. Figure 4 shows graphically
the results of such a computation for an Xy vibration as a function of
rotation about the X axis. It will be seen that at approximately
+ 35° and — 49° the Xy vibration will have a zero temperature
coefficient of frequency.

This curve has been checked experimentally, the check points being
indicated on the curve. Concentrating on a plate cut at -f- 35°,
which has been designated the ^7"-cut, it will be seen that this
type of plate offers considerable possibilities. Figure 5 shows the
frequency-temperature curves of a 2-megacycle AT-cxit plate and a

460

BELL SYSTEM TECHNICAL JOURNAL

standard F-cut of the same frequency and area. These curves not
only illustrate the reduction in temperature coefficient but also show-
that in the A T-cut plate the secondary frequency spectrum has been
eliminated over the temperature range of the test. This is to be
expected, for 35° is close to the 31° zero coupling point; hence such
coupling as does exist is small in magnitude.

y -20

o EXPERIMENTAL
CHECK POINTS

AC-CUT

-cut/

\at-cut

BC-CUT

■-80

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90

ANGLE OF ROTATION ABOUT X AXIS IN DEGREES (6)

Fig. 4 — Temperature coefficient of frequency of the vibration depending upon Cee' as a
function of rotation about the X axis.

These ^r-cut plates can be produced with a sufficiently low temper-
ature coefficient so that for most applications the temperature regu-
lating system can be discarded, and in addition a simplification of the
secondary frequency spectrum is obtained. Furthermore, the ad-
vantages of the F-cut plate, i.e., clamping and activity, have not
been sacrificed.

Additional tests on .4 T-cut plates indicate that it will be possible to
use them to control reasonable amounts of power without danger of
fracture. At 2 megacycles, 50-watt crystal oscillators would appear
to be practical and in some experimental circuits the power output
has been run up to 200 watts without fracturing the crystal. The

QUARTZ CRYSTAL CIRCUIT ELEMENTS

461

explanation for this lies in the fact that the reduction in magnitude of
the coupling to transverse vibrations has reduced the transverse
stresses which in the F-cut plate are responsible for the fractures.

Experimental crystals of this type have been produced in the
frequency range from 500 kc. to 20 megacycles. The possibility of
high frequencies, together with the elimination of the temperature

2500

2250

2000

1750

1500

1250

1000

750

500

250

-500

-750

-1000
-1250
-1500
-1750

-2000

Y-CUT/

AT- CUT (35°)

20 24

32 36 40 44 48 52 56 60 64
TEMPERATURE IN DEGREES CENTIGRADE

68 72 76

Fig. 5 — Frequency-temperature characteristics of AT-cut and Y-cut plates of same

frequency and area.
Frequency 1000 KC.
Dimensions:

Y-cut y = 1.970 mm. x = 38 mm. z = 38 mm.

AT-cut (35°) y = 1.675 mm. x = 38 mm. z' = 38 mm.

462

BELL SYSTEM TECHNICAL JOURNAL

control and the increase in the amount of power that may be controlled,
should result in a considerable simplification of future short wave
radio equipment.

APPENDIX

Elastic Equations

The general elastic equations for any crystal are given below, X\
Y' and Z' representing any orthogonal set of axes.

- XJ = cii'xx + Cii'yy + ciz'z/ + ci

— Yy = Ci2X/ + Cii'yy' + C23'z/ + C2

— Zz = Ci3 Xx + C03 yy -\- C33 Zz + C3^

— Y/ = CuX:c' + Cii'yy + Cn'z/ + C44

— ZJ = cx^lxj + ci'^yy + Ci^'zJ + C45

— Xy = Ciq'xJ + C2/yy' + Csa'z/ + C46

'y/

+ CyJzJ + Cie'Xj,"!

'y/

+ C^hZj + C^^'Xy

/y/

+ Css'Zx' + C3^Xy

/y/

+ C45'Zx' + Cui^Xy

'yj

+ ^55'2x' + Csfi'Xj,'

'y:

+ Cse'zx' + Ca^'Xy.

(3)

When in quartz X', F' and Z' coincide with the crystallographic
axes of the material {X the electric axis, Y the mechanical axis, and
Z the optic axis), equation (3) reduces to equation (2) of the text.
In addition the following relations exist between the constants of
equation (2) because of conditions of symmetry

Cl\ — C22, Cii — £"55, C66 — (cii -

Cii = — C23 = C56.

Cn)l2,

'^is — C23

The numerical values of these constants have been determined experi-
mentally by Voigt ^ and others.

cii = 85.1 X 10

C33 = 105.3 X 101

10 y* /-.„ = A Qi; v inio "y-

cm.^
dy.

cm.^

:cu = .6.95 X W

cn = 14.1 X IQi

Cii = 57.1 X W-^,cii = 16.8 X W
cm.''

r<-,6 = 39.1.

cm.''

_dyi

cm.^
dy^
cm.^

Using these constants it is possible to calculate the Ci/ for any orienta-
tion by means of transformation equations.^ The expressions giving
cu, C26.', ' • • Cqq (the constants relating to the Xy strain) in terms of
the Cij for rotation about the X axis, are given below, 6 being the

«W. Voigt, "Lehrbuch der Kristallphysik," 1928, p. 754.

^ A. E. H. Love, "Mathematical Theory of Elasticity," 4th ed., p. 43.

QUARTZ CRYSTAL CIRCUIT ELEMENTS

463

angle between the Z' and Z axis (Fig. 2).

CU — ^26 — <"3B — CH) — 0,

Cse' = Cii(cos- 6 — sin^ 0) + (cee — C44) sin 6 cos 0,
Cee' = C44 sin^ 0 + fee cos^ 6 — Icu sin 6 cos 0.

(4)

Piezo-Electric Equations
The inverse piezo-electric relations for the X' , Y', Z' system of axes
can be expressed by the following equations:

= di^'EJ + di^Ry + dzi'EJ
= dis'E/ + dizEy + dzi'E/
— d\i Ex -\- da Ey -\- dzi E^
= d,,'EJ + d2'JEy' + d,,-EJ
= die Ex' + di&'Ey' + ds&EJ.

(5)

When in quartz X', F', Z' coincide with the crystallographic axes,
eq. 5 reduces to the following:

Xx = d\\Ex

Jy = — dnEx

z, = 0

y^ = dnEx

Zx = — diiEy

(6)

where

^11 = - 6.36 X 10-

esu

dyne '

du = 1.69 X 10-8

esu
dyne

For rotation about the X axis,

d^Q — (dii sin d — 2dii cos 6) cos 6.

(7)

A Theory of Scanning and Its Relation to the Characteristics

of the Transmitted Signal in Telephotography and

Television

By PIERRE MERTZ and FRANK GRAY

By the use of a two-dimensional Fourier analysis of the transmitted
picture a theory of scanning is developed and the scanning system related
to the signal used for the transmission. On the basis of this theory a
number of conclusions can be drawn:

1 . The result of the complete process of transmission may be divided into
two parts, (a) a reproduction of the original picture with a blurring similar
to that caused in general by an optical system of only finite perfection, and
(6) the superposition on it of an extraneous pattern not present in the
original, but which is a function of both the original and the scanning
system.

2. Roughly half the frequency range occupied by the transmitted
signal is idle. Its frequency spectrum consists of alternating strong bands
and regions of weak energy. In the latter the signal energy reproducing
the original is at its weakest, and gives rise to the strongest part of the
extraneous pattern. In a television system these idle regions are several
hundred to several thousand cycles wide and have actually been used
experimentally as the transmission path for independent signaling channels,
without any visible effect on the received picture.

3. With respect to the blurring of the original all reasonable shapes of
aperture give about the same result when of equivalent size. The sizes
(along a given dimension) are determined as equivalent when the apertures
have the same radius of gyration (about a perpendicular axis in the plane
of the aperture).

4. With respect to extraneous patterns certain shapes of aperture are
better than others, but all apertures can be made to suppress them at the
expense of blurring. An aperture arrangement is presented which almost
completely eliminates extraneous pattern while about doubling the blurring
across the direction of scanning as compared with the usual square aperture.
From this and other examples the degradation caused by the extraneous
patterns is estimated.

TN the usual telephotographic or television systems the image field
•^ is scanned by moving a spot or elementary area along some recurring
geometrical path over this field. In the more common arrangement
this path consists simply of a series of successive parallel strips.
Imagining the path developed or straightened out (or in the more com-
mon case, the strips joined end to end), this method of scanning is
equivalent to transmitting the image in the form of a long narrow strip.
The theoretical treatment of such transmission has usually been
developed by completely ignoring variations in brightness across the
image strip, assuming the brightness to have a uniform distribution
across this strip. This permits the image to be analyzed as an ordin-
ary one-dimensional or single Fourier series (or integral) along the
length of the strip; and the theory is then developed in terms of the

464

A THEORY OF SCANNING 465

one-dimensional steady state Fourier components. Such a method of
treatment naturally gives no information in regard to the reproduction
or distortion of the detail in the original image across the direction of
scanning, nor, as will appear below, does it give any detailed informa-
tion in regard to the fine-structure distribution of energy over the
frequency range occupied by the signal.

The need of a more detailed theoretical treatment originally arose in
connection with studies of the reproduction of detail in telephoto-
graphic systems, especially in comparisons of distortion occurring
along the direction of scanning with that across this direction. Later,
this same need was strikingly shown by the discovery that a television
signal leaves certain parts of the frequency range relatively empty of
current components. Certain considerations indicated that a large
part of the energy of a signal might be located in bands at multiples of
the frequency of line scanning. Actual frequency analyses more than
confirmed this suspicion. The energy was found to be so closely con-
fined to such bands as to leave the regions between relatively empty of
signal energy.

Such bands and intervening empty regions are illustrated by the
examples of current-frequency curves in Fig. 1. These curves were
taken with the various subjects as indicated, and the television current
was generated by an apparatus scanning a field of view in 50 lines at a
rate of about 940 lines per second. The energy is grouped in bands at
multiples of 940 cycles and the regions between are substantially de-
void of current components. In addition to the bands shown by the
curves, it is known that similar bands occur up to about 18,000 cycles
and that there is also a band of energy extending up from about 20
cycles.

Certain of the relatively empty frequency regions were also investi-
gated by including a narrow band elimination filter in a television
circuit. The filter eliminated a band about 250 cycles wide and was
variable so that the band of elimination could be shifted along the
frequency scale at will. By shifting the region of elimination along in
this manner it was found that a band about 500 or 600 cycles wide
could be removed from a television channel between any two of the
current components without producing any detectable effect on the
reproduced image.

At a later date a 1500-cycle current suitable for synchronization was
introduced into a relatively empty frequency region, transmitted over
the same channel with a television current, and filtered out — all with-
out visibly affecting the image.

These results indicated quite clearly the need of a more complete

466

BELL SYSTEM TECHNICAL JOURNAL

A THEORY OF SCANNING 467

theory of the scanning processes used in telephotography and television
and led to the study outlined in the following pages. Since this study
will be confined to characteristics of the scanning processes all other
processes in the system, wherever used, will be assumed to be perfect
and cause no distortion.

The general trend of this more complete theory can be foreseen when
it is considered that to obtain an adequate reproduction of the original
it is necessary to scan with a large number of lines as compared with
the general pictorial complexity of this original. This means that for
any original presenting a large scale pattern (as distinguished from a
random granular background) the signal pattern along successive
scanning lines will, in general, differ by only small amounts. Thus, the
signal wave throughout a considerable number of scanning lines may
be represented to within a small error by a function periodic in the
scanning frequency. Since such a function, developed in a Fourier
series, is equal to the sum of sine waves having frequencies which are
harmonics of the scanning line frequency, it will be natural to expect
the total signal wave to have a large portion of its energy concentrated
in the regions of these harmonics.

Furthermore, the existence of signal energy at odd multiples of half
the scanning frequency will indicate the existence of a characteristic in
the picture which repeats itself in alternate scanning lines. It is to be
expected that such detail in a picture cannot be transmitted without
accurate registry between it and the scanning lines and that when the
detail spacing or direction or both differ somewhat from the scanning
line spacing and direction, beat patterns between the two will be pro-
duced in the received picture which may be strong enough to alter con-
siderably the reproduction of the original.

These phenomena are exactly what is observed, and will be treated
in more quantitative fashion in the discussion below.^

An Image Field as a Double Fourier Series

Let us first consider the usual expression of the image field as a
single Fourier series. The picture will be considered as a "still" so
that entire successive scannings are identical. Then if the long strip
corresponding to one scanning extends from — L to +L, the illumina-

1 In the following treatment an effort has been made to confine the necessary
mathematical demonstrations almost exclusively to two sections entitled, respec-
tively, "Effect of a Finite Aperture at the Transmitting Station," and "Reconstruc-
tion of the Image at the Receiving Station." Even in these sections a number of
conclusions are explained in text which do not require reading the mathematics if
the demonstrations are taken for granted. The occasional mathematical expressions
occurring in the earlier sections are very largely for the purpose of introducing
notation.

468 BELL SYSTEM TECHNICAL JOURNAL

tion £ as a function of the distance x along the strip may be expressed
as the sum of an infinite number of Fourier components, thus:

E{x) = £ a„ cos i -^ + <Pn] . (1)

In this summation a„ represents the intensity of the wth component
and (fn its phase angle. The complete array of these for all components
will vary if the picture is changed.

The cosine series above is very convenient for physical interpretation.
It will be simple, however, for some of the later mathematical work to
use the corresponding exponential series. The cosine series can be
returned to, each time> as physical interpretation is required. That is,
since

(TTX \

-7^ + <i5 ) = (ae^^)e(^'^^/'^> + (oe-i^)e(-»'^/^> (2)

the series in equation (1) can be written

+00

E(x) = X! Anexpiir(nx/L) (3)

ra= — 00

if we make
and

An = (l/2)a„exp (*>„)
A-n = (l/2)a„ exp (-*>„) (4)

and if we use the notation exp d = e^.

In this new summation the complex amplitude An represents both
the absolute intensity and the phase angle of the wth component.
The complex amplitude of the corresponding component with a nega-
tive subscript is merely the conjugate of this.

As has already been noted, however, and as might readily be ex-
pected, the single Fourier series in equations (1) or (3) above do not
always represent a two-dimensional picture with sufficient complete-
ness. In order to consider the two-dimensional field more in detail,
let us assume that Fig. 2 represents such an image field of dimensions
2a and 2b, and take axes of reference x and y as indicated. The
brightness or illumination of the field is a function E{x, y) of both x
and y. Along any horizontal line (i.e., in the x direction, constantly
keeping y = yi) the illumination may be expressed as a single Fourier
series

+ 00

E(x,yi) = XI Amexpi(nix/a). (5)

A THEORY OF SCANNING

469

Along any other line in the x direction a similar series holds with
different coefficients, that is, the ^'s are functions of y. They may

Fig. 2 — Scanned field and Image,
therefore each be written as a Fourier series along y

Am = 12 AmnexpiTr(ny/b). (6)

n=— OT

Substitution in equation (5) gives the double Fourier series,

E{x, y) = E E Amnexpiir I— +^)- (7)

For purposes of physical interpretation, as in the case of the simple
Fourier series, it is desirable to combine the -\-m, -\-n term with the
— m, —n term (giving the single (m, +w)th component) and similarly

470 BELL SYSTEM TECHNICAL JOURNAL

the +m, -w with the -m, -\-n terms (giving the single (m, -n)th com-
ponent). This brings equation (7) back to a cosine series,

nix ny .

(8)

00 +00

E{x,y) = Y. H CLrnn COS
m = 0 n= — 00

when

Amn = (l/2)a;„„exp (^V^n)
and

A-m-n = (l/2)a„„exp {-icpmn)

and where a„in is always a real quantity. Each term of this series
represents a real, two-dimensional, sinusoidal variation in brightness
extending across the image field. The image is built up of a superposi-
tion of a series of such waves extending across the field in various
directions and having various wave lengths.

Imagining brightness as a third dimension, we may, as an aid in
visualizing the components of an image field, draw separate examples
of various components as shown in Fig. 3. It will be noted that any
given component (m, n) passes through m periods along any horizontal
line in the image field, and through n periods along any vertical line.
The slope of the striations with respect to the x-axis is therefore
— mb/na (the negative reciprocal of the slope of the line of fastest
variation in brightness). For the same values of m and of n, the m, -\-n
component and the m, —n component have equal wave lengths but
are sloped in opposite directions to the x-axis. If m is zero the crests
are parallel to the x-axis; if n is zero they are parallel to the 3'-axis.
The component with both m and n zero is a uniform distribution of
brightness covering the entire image field. The wave length of a
component is

A complete array of the components, up to m and n equal to 4, is
illustrated in Fig. 4,

As of course is characteristic of the harmonic analysis, the wave
lengths and orientations of the components are seen to vary only with
the shape and size of the rectangular field, and to be independent of
the particular subject in the field. A change of subject, or motion of
the subject, merely alters the amplitudes of the components and shifts
their phase; but their wave length and inclination with respect to the
X-axis remain unchanged. Consequently, for the same rectangular
field all subjects appearing in it may be considered as built up from the
same set of components. For a "still" subject, the amplitudes and

A THEORY OF SCANNING

471

phase angles of the cosine components, or the complex amplitudes of
the exponential components, remain constant with time. For a mov-
ing subject these complex amplitudes may be considered modulated as
functions of time.

-3 COMPONENT

+ 2, -1-3 COMPONENT

0, +2 COMPONENT +2,0 COMPONENT

Fig. 3 — Examples of field components.

The real amplitudes amn for a circular area of uniform brightness on a
black background are relatively easily calculated, and this subject is
also a good one to study as a picture from some points of view because it
has a simple sharp border sloping in various directions. The amplitudes

472 BELL SYSTEM TECHNICAL JOURNAL

for a circle of unit illumination of radius R are

(9)

where /i is the first order Bessel function. In this particular subject
all components of a given wave length have equal amplitudes; and the

Mimm

Fig. 4 — Array of field components.

amplitudes may therefore be plotted as a function of wave length alone,
as in Fig. 5. The curve illustrates the rapidity with which the ampli-
tudes fall off for the higher order components in a subject of this nature.

A THEORY OF SCANNING

473

The Frequency Spectrum of the Signal

When an image field is scanned by a point aperture tracing across it,
each portion of the picture traversed causes variations in the light
reaching the light sensitive cell and is thus translated into a corre-

<^ 0.6

2 0.4

<,0.2

—

_/^

- >

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

WAVE LENGTH IN TERMS OF DIAMETER OF CIRCLE A.

Fig. 5 — Amplitudes of field components for a circular area of brightness.

spending signal. Further, each Fourier component in the field is trans-
lated into a corresponding Fourier component in the signal. An equiv-
alent translation occurs when a pencil of light traces over a photo-
graphic film in telephotography, or when a subject is scanned by a beam
of light in television, whether or not a simple flat two-dimensional image
is ever physically formed at the transmitting station. For clarity and
simplicity, the discussion will be confined to the case in which a point
aperture traces across a plane image field.

In most systems the aperture traces a line across the field and then
there is a sudden jump back to the beginning of the next succeeding
line. This discontinuous motion is naturally not easily subjected to
mathematical treatment. It is much simpler to deal with the equival-
ent result that would be obtained if the scanning point, instead of trac-
ing successive parallel paths across the same field, moved continuously
across a series of identical fields. Such an equivalent scanning motion
can fortunately easily be used because a double Fourier series represents
not only a single field, but a whole succession of identical image fields
covering the entire xy plane, and repeated periodically in both the x
and y direction as illustrated in Fig. 6.

474

BELL SYSTEM TECHNICAL JOURNAL

The equivalent of scanning a single field in parallel lines is obtained
by assuming that the scanning point moves across the repeated fields
along a sloping path as indicated. Let u be the velocity parallel to

Fig. 6 — Array of periodically recurring scanned fields.

the X axis and v the velocity parallel to the y axis. Then the picture
illumination at the scanning point at any instant, and consequently
the signal current, may be obtained by substituting

J\i Jo't'y

in the double Fourier series representing the image field, equation (8).
Of course the entire expression must be multiplied by a factor K which
is the constant ratio between the signal current and the picture
illumination. This gives for the real signal as a function of time "^

2 It will be noted that this process does not explore the picture completely, inas-
much as, no matter how fine the scanning, there will always be unexplored regions
between scanning lines. In this respect the process is quite analogous to that
followed in analyzing a function of a single variable into a simple Fourier series
when the values of the function are given only at discrete (even though closely spaced)
values of the variable. The complete exact theory, which necessarily depends upon
the size and shape of the finite scanning spot or aperture, will be given further below.

A THEORY OF SCANNING 475

OT +00

I{t) = KY. L O'mn COS
m=Q n= — oo

I mu , nv\ ^ , 1 ,^^.

Thus if u and v are constants, each wave of the image field gives rise
to a corresponding Fourier component of the signal. The frequencies
of the signal components are

^ 2a^2b ^^^^

The frequency spectrum of the signal is thus made up of a series of
possible discrete lines, the position of which in that spectrum is deter-
mined by u and v, that is, by the particular scanning motion employed.
We shall designate these lines by the indices m, -\-n and m, —n, as
they are correlated with the particular components of the image field
that generated them.

A different choice of values for u and v (so long as these, once having
been chosen, remain constant) changes the location of the lines in the
frequency spectrum, but their amplitudes, depending only on the
corresponding components of the image field, remain unchanged. In
other words, the lines in the frequency spectrum of the signal are
characteristic of the image field, and the scanning motion merely deter-
mines where they will appear in the frequency spectrum. Thus, if for
a given subject the distribution of energy over the frequency scale is
known for one method of scanning, it can be predicted for a great many
other methods.

To scan a field in lines approximately parallel to the x axis, the
velocity v must be made small compared to ii. Under such conditions,
u/{2a) of equation (11) is the line scanning frequency and v/{2b) is the
frequency of image repetitions (or "frame frequency"). The fre-
quency spectrum of the signal for a "still" picture thus consists of
certain fundamental components at multiples of the line scanning
frequency u/{2a), each of which is accompanied by a series of lines
spaced at equal successive intervals to either side of it. The spacing
between these satellites is the image repetition or frame frequency
v/(2b).

If the picture changes with time the amplitudes of these fundamental
lines and their satellites are modulated, also with respect to time. In
other words they each develop sidebands or become diffuse. The
diffuseness will not overlap from satellite to satellite unless the fre-
quency of modulation becomes as great as half the frame frequency.

476

BELL SYSTEM TECHNICAL JOURNAL

Thus for motions in the picture which are not too fast to be expected to
be reproduced with reasonable fideHty, this diffuseness of the funda-
mental lines and their satellites will not obliterate their identity.

A diagrammatic arrangement of some of the possible lines in a fre-
quency spectrum, with their corresponding m and n indices, is shown
in Fig. 7.

It is important to note that the correlation between the wave lengths
of the field components and the frequencies of the current components
is not the one that is naturally assumed on first consideration. We

••

O o

(- z

K Z

hi z

Q- <

UJ o

a to

,^ UJ

'i ?

5 -"

-t

z
(ij

i o

u.
o

REQUENCY
REQUENCY

o
a.

UJ
N

z
(ij

h. U.

1 1 . .1

1 1

1 1 ,

^.L±l

" + +
+ + +

o — OJ t*!

ro (M — o —

FREQUENCY

Fig. 7 — Diagram of signal frequency spectrum.

are quite likely to make the erroneous assumption that high frequencies
correspond to all sharp changes in brightness and that low frequencies
correspond only to slow changes. The error in this assumption is
readily realized by noting that sharp changes in brightness may gener-
ate very low frequencies if the scanning point passes over them in a
sloping direction. An actual correlation is shown schematically in
Fig. 8. It is seen that the same general type of correlation is repeated
periodically over the frequency scale at multiples of the line scanning
frequency. There are evidently numerous regions of the spectrum in
which short image waves, or fine grained details of the image field, may
appear in the signal. They are not confined to the high-frequency
region alone.

A THEORY OF SCANNING

477

0.2

1000 1500 2000 2500 3000 3500 4000 4500 5000

0.05

22,000

17,000 18,000 19,000 20,000 21,000

FREQUENCY IN CYCLES PER SECOND

Fig. 8 — Correlation between wavelength and frequency of signal components

In telephotography the frequency of line scanning is usually low and
the groups of lines in the frequency spectrum are so closely spaced that
such fine grained details of the signal are of little practical importance
as far as the electrical parts of the system are concerned. In television,
however, these bands are widely spaced, of the order of 1000 cycles or
so apart, and such details of the signal are quite important.

As a specific example, it is interesting to plot the frequency spectrum
of the television signal that results from scanning a circular area of
uniform brightness on a black background. So far as the present
theory extends, this may be done by converting the field components
of equation (9) into current components with the aid of equations (10)
and (11). Taking b/a = 1.28, the radius of the circle as b/3, and as-
suming that the field is scanned in 50 lines 20 times per second, we ob-
tain the amplitude-frequency spectrum shown in Fig. 9. Since it is
not convenient to show the individual current components — only 20
cycles apart — the curve shows simply the envelope of the peaks of
these components. At low frequencies, the energy is largely confined
to bands at multiples of the line scanning frequency, 1000 cycles, and
to an additional band extending up from zero frequency. In the re-
tions between the bands, the signal components are so small that they
do not show when plotted to the same scale. At higher frequencies the
signal energy as thus far computed is not confined to such bands. It

478

BELL SYSTEM TECHNICAL JOURNAL

AMPLITUDE

A THEORY OF SCANNING 479

will be shown farther on, however, that the effect of the use of a finite
aperture for scanning is to confine the signal energy more rigorously
to such bands throughout the frequency range.

The theoretical energy distribution for the circular area is in excel-
lent agreement with actual frequency analyses of television currents,
which show the energy confined to bands at multiples of the line scan-
ning frequency with apparently empty regions between. It is evident
from the theory so far, however, that these regions are not really empty
but are filled with weak signal components representing fine details of
the subjects; and subjects of greater pictorial complexity than a simple
circular area may be devised to give large signal components in such
regions. We must therefore look for other factors to explain why these
frequency regions do not transmit any appreciable details of an image.

Confusion in the Signal

With the usual method of scanning, one such factor is the confusion
of components in the signal. This confusion arises from the fact that
two or more image components sloping across the field in different
directions may intercept the line of scanning with their crests spaced
exactly the same distance apart along this line of scanning. As the
scanning point passes over them they thus give rise to signal current
components of exactly the same frequency. Consequently the two
image components are represented by a single, confused, signal current
component that can transmit no information whatever in regard to
their relative amplitudes and phases. This confusion evidently de-
pends on the scanning path.

If the image field is scanned in A'^ lines, the velocity v of the scanning
point parallel to the y axis is

and the signal frequencies from equation (11) are

/ = ^(»>+-«)- (13)

Field components with indices m, n and ni', n' such that

m -f ^ = m' -I- ^ (14)

give rise to current components of the same frequency.

480

BELL SYSTEM TECHNICAL JOURNAL

In other words, the bands of components in the frequency spectrum
really overlap. Consequently the components of one band may coin-
cide in frequency with the components of adjacent bands. Such coin-
ciding components are illustrated schematically in Fig. 10.

m BAND FREQUENCY m + 2BAND

Fig. 10 — Coinciding lines of confused bands.

It is obvious that a single a-c. component cannot transmit the sep-
arate amplitudes and phases of two or more image components. Con-
sequently the receiving apparatus has no information to judge how the
components in the original image are supposed to be distributed in the
reproduction.

The situation is most serious where the intensities of coinciding com-
ponents have the same order of magnitude, that is, at the centers of the
frequency regions intermediate to the strong bands. The confusion
in these regions is the most important factor that renders them in-
capable of transmitting any appreciably useful image detail.

On first consideration it would appear that the overlapping of bands
in the signal might result in a hopeless confusion. The situation is
saved, however, by the fact that components with large n numbers will
tend to be weak due to the convergence of the Fourier series, and are
further reduced, as will be shown later, by the effects of a scanning
aperture of finite size. They therefore do not usually seriously inter-
fere with the stronger components. The interference usually manifests
itself in the form of serrations on diagonal lines and occasional moire
effects in the received picture.

Confusion in the signal may be practically eliminated by using an
aperture of such a nature that it cuts off all components with n numbers
greater than N/2, that is, cuts off each band before it reaches the center
of the intervening frequency regions so that adjacent bands do not
overlap. The practical possibilities of this arrangement will be dis-
cussed further below.

The mere elimination of confusion in the signal itself does not neces-
sarily prevent the appearance of extraneous components in the repro-
duced image. The receiving apparatus itself must be so designed that

A THEORY OF SCANNING

481

when it reproduces all the image components represented by a given
signal component, it suitably suppresses all those but the dominant one
desired.

Effect of a Finite Aperture at the Transmitting Station

In the preceding pages the scanning aperture has been assumed as
infinitesimal in size, or merely a point. In any actual scanning system
the necessary finite size of the aperture introduces effects which will
now be considered for the transmitting end.

Let us first review briefly the usual theory of this effect when the
picture is analyzed simply as a one-dimensional Fourier series. Ac-
cording to equation (3) above, this series is

+ 00

Ei(x) = J^ AnexpiT{nx/L).

Let ^ be a coordinate fixed with respect to the scanning aperture as
shown in Fig. 11 and let the optical transmission of the aperture for

E (x)ORIGINAL PICTURE

Fig. 11 — Analysis of one-dimensional scanning operation. *

any value of ^ be T{^).^ Then if x is taken as a coordinate of the
origin of ^ the illumination at any point ^ of the aperture is

-foo

■Ei(» + ?) = L Anexpiirinlx + ^2/1),

(15)

3 This optical transmission may represent either the transparency of an aperture
of constant width or the width of an aperture which is a shaped hole in an opaque
screen.

482 BELL SYSTEM TECHNICAL JOURNAL

SO that the total flow of light through the aperture at any position x is

F,{x) = fm)E^(x + k)d^. (16)

•^aperture *

Since x is a constant with respect to the integration the exponential
term may be factored and the part involving x only may be brought
outside the integral sign. This gives

+00

Fi{x) = E Y{n)Anexpiir{nx/L), (17)

n= — 00

where

Y{n) = fn^) exp *7r(w^/L)^^ (17')

^ aperture

For a symmetrical aperture (that is, about the origin of |)

Y{n) = ^T{k) cos {irn^/L)d^ (17")

•''aperture

and Y{n) in this case is, therefore, a pure real quantity.

The important conclusion to be drawn from equation (17) as to the
effect of a finite transmitting aperture is that it multiplies the complex
amplitude ^„ of each original image component by a quantity Y{n)
which is independent of the picture being scanned. This is entirely
similar to the effect of a linear electrical network in a circuit, and the
quantity Y{n) is quite analogous to the transfer admittance of that
network.

The quantity Y{n) has been plotted for variously shaped apertures
in Fig. 12. For convenience in comparison, the ordinates of each curve
have been multiplied by a numerical factor to make F(0) = 1. The
curves show the characteristics that are by this time familiar, which
are that the effect of the finite size of the scanning aperture in the
transmitter is similar to that of introducing a low-pass filter in the
circuit, namely, cutting down the amplitudes of the signal components
for which n is numerically high, i.e., the high-frequency components.

The curves are remarkable, however, in that in the useful frequency
band (i.e. from w = 0 to something like half of the first root of Y{n)
= 0) all the distributions considered give practically the same transfer
admittance if the dimensions of the beam along the direction of scanning
are suitably chosen, as has been done in the figure. This results from

* The integral is mathematically taken from — co to -|- oo but the regions outside
the aperture give no contribution since the integrand is there equal to zero.

A THEORY OF SCANNING

483

CM
N
CM

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£. :D

isl rl
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IsT

-> II

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\

O / ' N

/ '^- —

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X 1 i

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\ /
1/

I-

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)}

8 z' :^ "

/ ?!8

/ ^ +

1 ^

1 (D :;: ^

i_ in

< -

.J "^

^Ol

c Li

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l.v

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O <

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D <

D C

0
3

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d

d d

EQUIVALENT TRANSFER ADMITTANCE Y(n)

484 BELL SYSTEM TECHNICAL JOURNAL

the physical Hmitation that the illumination in any part of the beam
must be positive, that is, the illumination from one part of the beam
must always add to that from another part and cannot subtract from
it.^ This observation enables one to define the resolution of two
apertures of different shapes as being equal along a certain direction
when their transfer admittances in the useful frequency range show
the same filtering effect, if that direction is used as the direction of
scanning. This will occur when the radii of gyration (about a normal
axis in the plane of each aperture) are equal. According to this
definition all the apertures illustrated in Fig. 12 have the same resolu-
tion along a horizontal direction.

When the picture is analyzed as a two-dimensional Fourier series
the equations which have been given above become

Ei{x, y)=22 zZ Amn exp nr[ r -7- ) >

Ei{x + k,y + ri)

= E L ^^™exp«x — !--- - + -^^ ^ ' (18)

Fiix, y)= f fr^a, v)E,{x -{- ^,y + v)d^drj, (19)

•^ •^aperture

Fi{x, y) = Z Z Y,(m, n)Amn exp tV f — -f ^ \ , (20)
where

Yi(m, n)= f fr^i^, r?) exp ^tt ( ^ -f ^ ) d^dr,. (20')

ty t^ aperture \ ^ '' /

For an aperture symmetrical about both | and 77 axes

Fi(m, n) = f fr.a, v) cos ^('^-\-^) d^drj. (20")

*J ^'aperture \ ^ '^ '

^ The shape of the transfer admittance curve near n = 0 depends upon the power
of n in the first variable term of the Taylor expansion for Y{n) about n = 0, and
upon the sign of this term. Assuming a symmetrical aperture, the expansion from
equation (17") is

Y{n) = fTdi - ^ fe-Td^ + ^ f^Td^ .

Since T is everywhere positive the first variable term is always in ti' and negative.
The shape of the curve near w = 0 is, therefore, always a parabola (indicated in Fig.
12), which can be made the same parabola by suitably choosing the two disposable
constants in the aperture. Even after departing from this common parabola, the
curves maintain the same general shape over a substantial range; for the next variable
term is in n* and positive, and has the same order of magnitude for all usual types
of apertures. Consequently, the curves for these apertures have approximately
the same shape over a wide range extending uj) from n = 0. The results are the
same for an unsymmetrical aperture, but the reasoning is more involved.

A THEORY OF SCANNING 485

In the two-dimensional case T(^, 77) is defined, for a hole in an
opaque screen, as unity throughout the area of the hole, and zero for
the screen. Where the aperture is covered with a non-uniform screen
T may take on intermediate values.

The transfer admittances have been calculated for a variety of
shapes of aperture in Appendix I. It will be noted that for those types
of aperture for which T can be separated into two factors, one a func-
tion of ^ only and the other a function of 77 only, namely, for which

TiU, 77) = rj(^) • T,(r,), (21)

then equation (20') becomes

Yi{-m, n) = I T^(^) exp {iTrm^/a)d^ 1 ^,,(17) exp (iirnr]/a)dr]

»^aperture •-'aperture

= Y^im) . Y,(n) (22)

and Fj and F, are each one-dimensional integrals of the type illustrated
in Fig. 12.

The rectangular aperture is a simple case of this type. Assume
the field to be scanned in N lines and take the dimensions of the aper-
ture, 2c and 2d parallel to the x and y axes, respectively, as

Then

,, / s sin Trmc/a
Y^{m) =

and

wmc/a

sin TTfid/b
■wnd/h

and the frequency corresponding to a given signal component mn is,
from equation (11)

Thus, Yi{m, n) considered as a function of the signaling frequency
corresponding to each component of indices mn, consists of a succes-
sion of similar curves u/2a cycles apart, corresponding to the successive
integral values of m (these curves are themselves really not continuous
but consist of a succession of points u/{2aN) cycles apart. For con-
venience, however, the drawings will always show the curves as con-

486

BELL SYSTEM TECHNICAL JOURNAL

m = 45
n=o

m=46

y\M

45 46

FREQUENCy

Fig. 13a — Detail of equivalent transfer admittance of aperture for two-dimensional

scanning.

A THEORY OF SCANNING 487

tinuous). Each of the curves is of the equation

. 2irNc I - mu
sin — ^ W ■" ^
Y\{m, n) = Fj(m)

u \ 2a j

and therefore has a peak of the value Y^(m) at the point where n = 0
or/ = mu/2a, and trails ofif from the peak in each direction according
to a curve of the same shape as curve "A" In Fig. 12. The successive
curves are all of identical shape, but each one is to a reduced scale of
ordinates as compared with the preceding (In the useful frequency
range) as Imposed by the factor Y^im).

The peaks, it will be noted, occur at the frequencies occupied by
what have been called the fundamental components (as distinguished
from the satellite lines) In the discussion above on the frequency spec-
trum of the signal.

Assuming N to be 100 and for simplicity taking the factor u/2a as
equal to 1, a plot is shown in Fig. 13a of Yi{m, n) over a very limited
region near the upper end of the useful frequency range. The curve
shown In a solid line represents Fi(m, n) for m = 45, and the dotted
curves on either side represent the function for m = 44 and 46,
respectively.

The function has been redrawn for the complete useful range of
frequencies and a little beyond, in Fig. 13b, with the frequencies to a
logarithmic scale. This logarithmic plot opens out the scale at the low
frequencies and enables the fine structure of the function to be indi-
cated there, and still enables the complete range of useful frequencies
to be shown without requiring a prohibitive size of drawing (it has,
however, the disadvantages that the distortion in the frequency scale
then masks the symmetry of the individual curves around the funda-
mental lines, the similarity of shape of these individual curves, and
also the constant frequency separation between the successive funda-
mental lines).

The function Fi(m, n), as Is clear from equation (22) and Figs. 13a
and b, consists of a sort of envelope function Fj(w), "modulated" by
a fine structure function Y,,(n). The latter function has the value
unity at the positions of the fundamental lines in the frequency spec-
trum of Fig. 7 and diminishes for the satellite lines in the same way
that the envelope function diminishes for the fundamental lines
away from zero frequency. It will be seen that the envelope function is
the only one obtained by the simple one-dimensional analysis. The

488

BELL SYSTEM TECHNICAL JOURNAL

1 1

[

ju>

<zZ

o
o

E-,

^^-

o
in
II

_^,

--""^

__-

^^—'

,--''

^■

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/
/

,' '

2 —

(

ro
II

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E
— o

^_^

= —

— =

(T^

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)

t\JO

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o8
El

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_.^

— — ^

^>^

)

—

--^'^-

--.^^

/oo

C f- "

n Ec

EQUIVALENT TRANSFER ADMITTANCE Y(m,n)

A THEORY OF SCANNING 489

complete function shows, by the very small transfer admittance in the
regions half-way between the fundamental lines, an additional reason
why the signal currents in these regions will be weak and relatively in-
capable of transmitting appreciable image detail.

Examination of the other apertures for which computations are given
in Appendix I will show that, in general, for all ordinary apertures the
same broad phenomena are observed as for the rectangular aperture,
although it is not always possible to express the complete function in
the simple product form above, in which case the curves for the suc-
cessive values of m will vary gradually in shape.

The final signal current is proportional to the light flux through the
aperture, given in equation (20). Neglecting constant factors it may,
therefore, be written as

m = L E Fi(m,nM.„expiVf— +^)/. (24)

Reconstruction of the Image at the Receiving Station

At the receiving station the signal current is translated back into
light to illuminate an aperture moving in synchronism with the one
at the sending end. Neglecting constant factors the flow of light F^it)
to the receiving aperture is

^2(0 = m- (25)

Let Ez(x, y) be the resulting apparent illumination (integrated with
respect to time) at a point x, y of the reproduced image, or, in tele-
photography, the integrated exposure of the recording film at this point.
This illumination may be expressed as a double Fourier series, similar
to equation (7) (but primed subscripts will be used to distinguish them
from those of that equation).

where

Mx, y) = L L 5».'.' exp iV ( — + ^ y (26)

Bm'n' =^^ j'^" £^E,(x,y)exp -i^ (^H^ ^I^yxdy. (27)

Reproduction of detail in the image may be studied by comparing
these components with the corresponding ones of the original image.

The apparent illumination is the same as if the aperture traced a
single strip across repeated fields in the xy plane as illustrated in Fig.
14, and all of the repeated fields included between y = — b and

490

BELL SYSTEM TECHNICAL JOURNAL

y = -^ h were cut out and superposed to form the image. Let
Ez{x, y) be the illumination of this strip. Then, since the exponential

-T]

-a

- '

2a 1 1 3a

Y 1 t

-I

-b

Fig. 14 — -Analysis of received picture.

factor of the integrand in equation (27) is periodic in x and identically
reproduced in each of the fields, the integral is equal to

Bm'n' = -T—u I I ^ii^^ y) exp -«V ( — - + —r-] dxdy. (28)

The limits —b to -\-b in y and the infinite limits in x may be used be-
cause the illumination is zero everywhere outside of the strip.

Again taking a coordinate system ^-q fixed with respect to the aper-
ture, such that

x = ^ + ut, y = 7] + vt (29)

the instantaneous illumination of any point covered by the aperture is,
neglecting constant factors,

T2{^, v)I{t) = T^ix - ut,y - vt)I(t). (30)

The total illumination of any point xy in the image strip is thus

Esix, y) = I T2(x — ut,y — vt)I{t)dt.

Substitution in integral (28) and a change in the order of integration

A THEORY OF SCANNING

491

gives

J /»+» /•+* /•+<»

B„,'n' = 4-T I I I ^2(^ - tit, y - Vt)I{t)

. ( m'x , n'y\ , , , ,^,.
• exp —iTT ( 1 — -^ 1 dxdydt. (31)

Changing to the ^r? system

1 r+" r"-"' f"^*^,, s^,,v . ( m'u , n'v\ ^

ty— 00 ty —b—vi »■' —00 ^ '

exp -zV ( ^ + ^ ) ^^Jrjff/. (32)

This integral may be considered as the surface integral of a function
(^(?7, /) taken over a strip shaped area shown in Fig. 15, in elements of

^^"vcb

b
k

n^^b

"\^

-b
k

-J

I '^^^

Fig. 15 — Equivalent integration regions.

the type indicated as /. From this it may be seen that, when also
integrated in elements of the type indicated as //,

I <p{r], f)dr]dt = ( I <p{t], t)dtdr}.

x> J—b—it tJ—00 *J (—b—ri)jv

Consequently

i^^)

■l^m'n'

4ab

. I m II , nv , ,

I T2a, v)m exp

00 J( — b—ri)IV •/— 00

• exp -«V ( ^ + ^ ) d^dtdn. (34)

Consider now the intensity Bm'n' of a final reproduced picture com-
ponent m' , n' resulting from a single component m, n in the signal as

492 BELL SYSTEM TECHNICAL JOURNAL

expressed by equation (24). The Integral becomes

■Dm'n'

4ab

. / m — m' , n — n' , ,
exp 47r I w H 7 — z; ) t

' exp -iTr(^ + ^\ d^dtdr]. (35)

It will be noted that the exponential function of / is periodic in t, one
of the periods being to = 2b /v. Furthermore, this is just the difiference
between the upper and lower limits in /. Hence the integral in / may
be written

I exp i

tir I II -\ ; V ) tat.

a 0

This integral is zero except when

ni — m' , n — n' . .^^,

u -\ r — y = 0, (36)

a 0

in which case

I = to. (37)

The meaning of these last few equations is clear. It is, as would
be expected, that a signal component m, n does not give rise to all
components m', n' in the final received picture, but that these latter
components are in general zero unless m' and n' satisfy a definite rela-
tionship with m and n, expressed by equation (36). A somewhat un-
expected result is, however, that equation (36) allows some other
w', w' components besides the normal one for which w' = mandn' = n.
That is to say, a given signal component m, n in the line will reproduce
in the final picture not only a corresponding m, n component, but as has
been foreshadowed in the discussion on confusion in the signal, it will
also reproduce certain other components with different indices.

Let us consider first, however, the reproduction of the normal
component for which m' = m and n' = n, which is obviously allowed
by equation (36). The amplitude Bmn is then, neglecting constant
factors,^

Bmn = A„,nYi(m, fi) Yi{m, n), (38)

where

F2(m, n) = r^ r^ T,{^, r,) exp -iV ( ^ + ^ ) d^dr,. (38')

^ The constant factor neglected as compared with equation (35) is to/i'iab). The
/o is the period of image repetitions (or "frame period"). It appears here because
the brightness of a single image depends on how quickly it is reproduced.

A THEORY OF SCANNING 493

The quantity Fa it will be noted is almost the same, for the receiving
aperture, as the Yi is in equation (20') for the sending aperture.
Thus, on the normally reproduced component the receiving aperture
merely adds whatever filtering action it has to that which has already
been caused by the sending aperture.

As noted, in addition to this normal component, the integral (35)
exists in general for other values of m' and n' and thus gives rise to
extraneous components in the reproduced image. If equation (36) is
applied particularly to the usual system of scanning in N lines in
which as in equation (12), v = uh/{Na), it becomes

m+^=m'+^. (39)

For values of m' and n' satisfying equation (39), the reproduced com-
ponent has the complex amplitude (neglecting constant real factors)

Bm'r.' = AmnYrim, u) Yi{m' , n'). (40)

Looking back at equation (14) and comparing it with equation (39)
it may be seen that these components correspond in indices to the
original image components that are confused in the signal to give only
one signal component. The result is, therefore, after all quite reason-
able from a physical point of view. For when a signal of a certain
frequency is transmitted over the line the receiving apparatus has no
information by which to judge which component in the original picture
it is supposed to represent. So, as shown by equation (40) it impar-
tially reproduces every one of the components it could possibly repre-
sent, each component with the intensity and phase it would have if it
were really the one intended to be represented by the signal. The
components are then all superimposed in the picture.

From this development it is clear that the process of scanning an
image field in strips and reproducing it in a similar manner not only
reproduces the components of the original image but also introduces
extraneous components. The reproduced field thus consists of two
superposed fields: a normal image built up from the normally repro-
duced components, and an additional field of extraneous components.
Although not really independent, it is convenient to consider these
two fields as existing separately, and thus to think of the normal image
field as having an extraneous field superposed on it.

Considering the normal field alone, we may term the reproduction of
its detail as the reproduction of normal detail. There is a loss in such
reproduction, for both the transmitting and receiving apertures intro-

494 BELL SYSTEM TECHNICAL JOURNAL

duce a relative loss in the reproduction of the shorter wave components.
Consequently there is a loss of definition in the finer grained details of
the normal image. This type of distortion due to aperture loss may be
termed simple omission of detail.

In addition to the simple omission of detail, the normal image is
masked by the presence of the extraneous field. The more pronounced
features of this field are the line structure and serrated edges that it
superposes on the normal image. Its presence is not only displeasing,
but it also masks the normal image components and thus results in a
further loss of useful detail. This type of loss may be termed a
masking of detail or a masking loss. It is true that the extraneous com-
ponents may sometimes give rise to an illusory increase in resolution
across the direction of scanning in special cases where they add on to
the diminished normal components in just the right phase and magni-
tude to bring the latter back to their phases and intensities in the
original image, giving no resultant distortion whatever. (In all such
cases, however, to obtain this benefit it is necessary to effect a quite
accurate register between the original image and the scanning lines or
the distortion is very large. Such accurate registering is generally
impractical and may be definitely impossible if the registry required for
one portion of the image conflicts with that required in another portion.
Such cases may, therefore, in general be disregarded.)

The Reproduction of Normal Detail

The preceding theory permits a numerical calculation of the repro-
duction of detail in the normal image. This is given directly by equa-
tion (38) above.

In order to make some of the discussion in the following pages more
concrete and specific the sending and receiving apertures will be taken
alike; this condition, therefore, gives [F(m, w)]^ as a measure of how
well the various components are reproduced. If a picture be assumed
in which all the original components have the same amplitude then
\_Y{m, n)'J' is the amplitude of the reproduced normal components.

The relative admittance for any given pair of apertures may be
calculated from equations (20') or (38'). Such calculations have been
made for various apertures and the results summarized in Appendix II.

The admittance of an aperture is not in general uniquely determined
by the wave length of a component, but also depends on the orientation
of the component with respect to the aperture. The admittances of
reasonably shaped apertures do, however, decrease in general with
increasing numerical values of the indices m and n ; and the shorter wave
components are, therefore, in general, less faithfully reproduced than
the longer wave ones.

A THEORY OF SCANNING

495

A circular aperture furnishes a simple example of such reproduction —
because its admittance, from its symmetrical shape, is a unique func-
tion of the wave length of a component. In other words a circular
aperture reproduces normal detail equally well in all directions. We
may, therefore, simply plot \^Y(m, n)2^ as a function of the component
wave length as in Fig. 16, and this single curve is a measure of how

^ 0.8

§ 0.6
a:

u.
10

5 04

— —

1-
1-

§ 0.2
$
a
0

. ^ ^

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

WAVE LENGTH IN TERMS OF DIAMETER OF APERTURE Sh-

Fig. 16 — Equivalent transfer admittance for circular apertures at both sending and
receiving ends, vs. wavelength.

well the various normal components are reproduced. The shorter
wave components are practically omitted in the reproduction of an
image.

Other apertures do not reproduce normal image detail equally well
in all directions because their admittances depend on the slope of a
component. To simplify the consideration of such apertures we may
resort to a practice commonly used in discussing telephotographic or
television systems, and that is, we may take the resolution along the
direction of scanning and across the direction of scanning separately as
criteria of their performance.

Neglecting the small slope of scanning lines with respect to the
X axis of the image field, the admittance of an aperture for components
normal to the direction of scanning is Y(m, 0). Consequently, we
may take [F(m, 0)]^ as a measure of the reproduction of normal detail
along the direction of scanning. In a similar manner we may take
[[F(0, m)J^ as a measure of the reproduction of normal detail across the
direction of scanning.

It thus follows that an aperture gives the same resolution of normal
detail along the direction of scanning and across the direction of scan-

496

BELL SYSTEM TECHNICAL JOURNAL

ning when the two admittances F(m, 0) and F(0, n) are substantially
equal for components of the same wave length over the useful range.
Circular apertures, square apertures and other apertures that are
suitably symmetrical fulfill this condition exactly, and consequently
give equal resolution of normal detail in the two directions.

The curve C^(0> ^)!]^ has been plotted, by way of illustration for a
rectangular aperture, in the middle line of Fig. 17.

SECOND
EXTRANEOUS PATTERN

n'=n-2N

-=^' k- ^'.'

Y(o,n)^^ /YCo,n')

s /

FIRST
EXTRANEOUS PATTERN

n' = n-N

Y(o,n)\XYfo,n) N

/'Y(p,n') \(^

^xY(o,n)

FIRST
EXTRANEOUS •

PATTERN

n' = n+N y

y SECOND <^°'^')
EXTRANEOUS ^s
PATTERN \

n' = n+2N

N ■^-IZU--''' 2N

Fig. 17 — Reproduction of original and extraneous patterns.

It may be noted incidentally that the simple omission of detail which
occurs in the reproduction of normal components is quite similar to the
loss of resolution that an image suffers when it is reproduced through

A THEORY OF SCANNING 497

an imperfect optical system. Specifically the effect of a sending or
receiving circular aperture alone, or Y{ni, n), is the same as that caused
by an optical system which reproduces a mathematical point in the
original as a circle of uniform illumination (circle of confusion) in the
image of the same size (with respect to the image) as the scanning
aperture. The effect of the two apertures in tandem, or [F(w, w)3^,
may be very closely simulated by a circle of confusion of about twice
the area of either aperture, as can be judged from the discussion which
has been given above regarding the curves in Fig. 12.

The Extraneous Components

It will be clearly understood that the discussion immediately pre-
ceding has been confined entirely to the normal image components,
that is, to the image that would be seen if no extraneous components
were present. In particular, it should be clear that the reproduction
of normal detail equally well in the direction of scanning and across the
direction of scanning does not mean that the details of the total result-
ant image will be seen equally well in the two directions, for the
extraneous components will to a certain extent mask the normal
image.

In the same manner as for the normally reproduced components, the
amplitudes of the extraneous components, according to the preceding
theory, are given by equation (40) above, where

m' = m + ju,
n' = n — txN,
where

M = an integer = m' — m = (\/N){n — n'). (41)

The composite transfer admittance F(w, n) • Y{m', n') may therefore
be taken as a measure of the extent to which the extraneous components
are introduced. If a picture be assumed in which all the original com-
ponents have the same amplitude then Y{m, n) • Y{ni', n') is the
amplitude of the extraneous components.

A given original component of indices m, n gives rise to a whole
series of extraneous components, ni' , n' , as ^l ranges from 1 up through
the positive integers and — 1 down through the negative integers.
As an illustration we have plotted the case of a rectangular aperture
of a width just equal to the scanning pitch, in Fig. 17, which has just
been referred to in considering the normal components. The two
lines marked "first extraneous pattern" show the relative amplitudes
for /i equal to 1 and — 1, respectively, and those marked "second extran-

498 BELL SYSTEM TECHNICAL JOURNAL

eous pattern," for /x equal to 2 and — 2, respectively, for m = 0. (The
shift from m' = 0 to m' = ±1 and ± 2 has been ignored since if N is
at all large this has a negligible effect on Yim', n'), as may be noted from
Fig. 13.) An examination of Fig. 17 and a consideration of the nature
of Y(m, n) and Y{m' , n') shows that the principal interference effect
will come from the pattern for which |m| = 1, and that the relative
amplitudes become very small as /i increases in absolute magnitude.
In general, therefore, only the first extraneous pattern may be con-
sidered as of really serious importance. Considering this pattern in
Fig. 17 it will be seen that the amplitude F(0, n) F(0, n') increases as
\n'\ increases from zero, the extraneous components becoming more
and more comparable to the normal components. At N/2, both com-
ponents are of the same amplitude, and the extraneous components are
therefore masking the normal components. It will be noted that the
index region at N /2 corresponds to the centers of the relatively empty
regions in the frequency spectrum of the signal. The large masking
effect caused by the extraneous components explains why such small
signal energy as exists in these regions is almost completely incapable
of transmitting any useful image detail.

It will be noted that the components with values of \n'\ in the
neighborhood of N/2 and greater are in general almost parallel to the
direction of scanning. The masking loss will therefore be greatest
across the direction of scanning and practically negligible along the
direction of scanning. This is quite reasonable because the extraneous
components constitute the line structure of the reproduced image, and
should therefore cause the greatest loss of detail across the direction of
scanning.

For clarity in the explanation up to this point, masking loss has been
discussed as if an extraneous component could only mask the normal
component with which its indices happened to coincide. In reality the
masking is of a more serious nature. An extraneous component un-
doubtedly obscures any normal component that has about the same
wave length and the same slope across the field even though it does not
exactly coincide in these characteristics.

More detailed curves than Fig. 17, showing the amplitudes of the
extraneous components have been prepared in Appendix II. These
also show the results for other index values of m than zero, and for
other than the simple rectangular aperture. The results indicate that
the extraneous patterns diminish in intensity progressively as more
overlap is tolerated between adjacent scanning lines, at the expense, of
course, of increased aperture loss for the normal components. This
point will be taken up again below.

A THEORY OF SCANNING

499

The reality of these extraneous components is strikingly demon-
strated in Fig. 18, for which we are indebted to Mr. E. F. Kingsbury.

a. Original. b. Transmitted.

Fig. 18 — Fresnel zone plate.

This shows at (a) the original of a Fresnel zone plate and at (b) the
picture after transmission through a telephotographic system. The
first extraneous pattern is very prominent in the lower corner of (b)
and a detailed study of the slope and spacing of the extraneous striations
shows them to be in exact accord with the theory which has been given.''
The special case of the extraneous components which are formed when
the original consists of a flat field is of some interest due to the high
visibility of these components under such a condition. This scanning
line structure is quite familiar as an imperfection in many pictures

^ The extraneous pattern, although it is (and should be according to the theory)
very nearly a transposed reproduction of the original pattern, must not be confused
with a long delayed echo of that original pattern. In other words, if only the lower
half instead of the whole of (a) had been transmitted, the lower half of {b) would
still have been exactly as it is, the extraneous components being generated entirely
irrespective of whether components representing a similar configuration exist in
other portions of the original or not.

In the region about half-way between the centers of the normal picture and the
first extraneous picture the resulting pattern gives very much the appearance of
another set of extraneous components. It is not such, however, that successive
rings are not really bright and dark, as they would be in the case of a genuine ex-
traneous component, but alternating uniform gray and striped black and white,
so that the average intensity along the circumference of a ring is independent of the
diameter of the ring, except for some photographic non-linearity.

500 BELL SYSTEM TECHNICAL JOURNAL

transmitted by telephotography and television. It can be removed
only by insuring that Yirn' , n') shall vanish whenever n' = N, so that

F(0, 0) • Y(fx, fxN) = 0.

The requirement can be met for the elementary shapes of apertures
A, E and Foi Fig. 12, but cannot be met in the others. In these other
cases the overlap between adjacent scanning lines is usually adjusted
so that the requirement is met for fx = ± 1, to remove the most serious
pattern. Thus, for example, for the circular aperture B this requires
an overlap of around 25 per cent.

The Reproduction of Detail

In optical instruments the reproduction of detail is usually measured
by what is called the "resolving power" which in turn is defined from
the smallest separation between two mathematical point (or parallel
line) sources of light in the original which can be distinguished as
double in the reproduced image.

For the present it is perhaps simpler to consider another criterion of
the resolving power, namely, the shortest element length in an image
resembling a telegraph signal, used as an original, which can be recog-
nizably reproduced with certainty in the received picture. For reasons
that have already been mentioned above it is necessary to insist that
the received picture be recognizable with certainty without any registry
requirement between the original image and the scanning lines.

Using this criterion for the resolution along the direction of scanning
and assuming the apertures at the sending and receiving ends to be
rectangular and of the same length with respect to the picture size,
the minimum signal element required for a recognizable picture (as set
by the apertures as distinguished from the electrical transmission
circuits) will be of about the length of either aperture. For other
shapes of aperture the minimum element length will be very nearly the
length of the equivalent rectangular aperture using the term "equival-
ent" in the same sense that it was used in the discussion regarding
Fig. 12.

According to the same criterion, for the resolution across the direc-
tion of scanning the minimum element length required for recognizable
transmission, in the case of a rectangular aperture of width equal to the
scanning pitch, will be twice the scanning pitch. It will be noted that
this is twice the length which would be required if only the normal im-
age components were reproduced, and this difference may be considered
as a measure of the degradation caused by the masking effect of the
extraneous components for this arrangement of apertures.

A THEORY OF SCANNING 501

This figure for the degradation must be taken with a certain reserve,
partly because the exact telegraph theory for the criterion of resolution
considered has really been inferred rather than presented in complete
logical form, and partly because the figure may be expected to vary
according to the criterion of resolution chosen. Some rough studies
have indicated the degradation to be materially less if the more con-
ventional criterion of resolution (two parallel line sources of light) were
used.

This degradation may be estimated in another manner. In Ap-
pendix II the extraneous components have been computed for a variety
of apertures and degrees of overlap between adjacent scanning lines.
In Fig. 19 there have been plotted the maximum amplitudes of these

APERTURES

f<^

! 1.5 2.0 2.5 3.0 3.5 4.0

BLURRING RELATIVE TO SQUARE APERTURE OF SCANNING PITCH WIDTH

Fig. 19 — Magnitude of extraneous components as a function of resolution.

extraneous components in each case (the first and second extraneous
patterns being plotted separately) as a function of the relative coarse-
ness of resolution for the normal image alone. This latter quantity is
taken relative to a rectangular aperture of width equal to the scanning
pitch, and, for example, for a rectangular aperture of width equal to
twice the scanning pitch, is represented by the figure 2. For conveni-
ence, above the various points have been inserted small diagrammatic
representations of the corresponding apertures. Also for convenience
the points have been arbitrarily connected together.

From inspection of Fig. 19 several conclusions may be drawn,
namely,

502 BELL SYSTEM TECHNICAL JOURNAL

1. Considering apertures of a given shape, the more overlap allowed
between adjacent scanning lines the weaker will be the extraneous pat-
terns but the coarser will be the reproducible detail in the normal
image.

2. Not all shapes of aperture are equally efficient in suppressing
extraneous components, and at the same time retaining a given resolu-
tion of normal detail. Of the shapes considered, the rectangular
aperture is least efficient in this respect, and the full-wave sinusoidal
aperture {E in Fig. 12), is the most efficient.

3. Although not proved, it may be inferred from the figure that the
finest resolution in the normal image that can be obtained (assuming
a given scanning pitch) without showing a first order extraneous pat-
tern on a flat field, is that obtained with the rectangular aperture of
width equal to the scanning pitch.

4. With the most suitable aperture it is possible practically to sup-
press the extraneous components, at the expense of coarsening the
normal reproducible detail to slightly under twice that given by the
rectangular aperture just mentioned.

The last point in particular enables us to draw a conclusion in regard
to the degradation contributed by the extraneous components. For a
rectangular aperture of width equal to the scanning pitch it appears
that the degradation amounts to a little less than doubling the coarse-
ness of resolution to normal detail. This substantially checks the
estimate which has already been made above. It may further be sur-
mised for all the other shapes of aperture shown with a value of abscissa
under 2 that as the degradation contributed by the extraneous com-
ponents is reduced, the coarseness of resolution to normal detail is in-
creased to just about make up for this, and that in the overall picture
the minimum element length which can be recognizably reproduced
remains substantially constant at about twice the scanning pitch.^ For
aperture arrangements with values of abscissa over 2, either the ineffi-
ciency in suppressing extraneous components, or the unnecessarily
large overlap, tends to coarsen the overall resolution to a minimum
elementary length greater than twice the scanning pitch. In this
region the line connecting the points has been dotted.

* It may very well be that even if all these aperture arrangements transmit an
about equal amount of information they do not give the same psychological satis-
faction to the viewer at the receiving end. The general effect of a square aperture
of scanning pitch width is to give a "snappy" appearance, disturbed, however, by
the presence of the extraneous patterns. When these are removed, keeping the over-
all resolution about the same, the appearance becomes "woolly" or "fuzzy."

A THEORY OF SCANNING 503

An Estimate of the Idle Frequency Regions

As mentioned at the beginning of this paper, the frequency regions
between the strong bands appear to be empty when examined with a
frequency analyzer of limited level range, or when a narrow band
elimination filter is used in connection with visual observations of the
reproduced image. These regions are not really completely empty,
but do contain weak signal components as shown by the preceding
theory, which are not, however, particularly useful inasmuch as, in
the final result, they give rise about equally to components simulating
the original picture and to masking extraneous components. The
regions may, therefore, be considered as idle.

The factors determining the extent of these idle regions are too com-
plicated to permit an exact theoretical evaluation of their width, but
an estimate may be attempted from an inspection of Fig. 12 and of the
curves given in Appendix II.

From Fig. 12 and the experience that along the direction of scanning
the minimum recognizably transmitted elementary signal length is the
length of a rectangular aperture it can be deduced that in the absence
of extraneous components the useful band of an aperture extends up to
the point where its relative admittance, for a single aperture, is in the
neighborhood of 0.65. For two apertures in tandem the corresponding
relative admittance is 0.65^ = 0.42.

Now in Fig. 27 of Appendix II the extraneous components are very
small and may be considered negligible. According to the above cri-
terion, therefore, the useful frequency band constitutes approximately
54 per cent of the total space. The idle frequency regions would,
therefore, occupy the remainder, or 46 per cent of the total space.

Experimental examination of a television signal with a narrow band
elimination filter gave the width of the idle regions as 50 to 60 per cent
of the total space. This was for a field scanned with a circular aperture
giving a one-quarter overlap of scanning strips. The discrepancy for a
quantity so vaguely defined is not large but is probably due to incom-
plete utilization of even the theoretically active region by the television
set because of inherent imperfections in parts of the complete system
outside the scanning mechanism proper.

The width of the individual idle bands is then about half the fre-
quency of repetition of scanning lines. For most systems of telephotog-
raphy this runs in the order of magnitude of one cycle per second, mak-
ing the waste regions very narrow and close together. For systems of
television the waste bands come in much more significant "slices," al-
though the same fraction of the frequency space is wasted. For ex-

504 BELL SYSTEM TECHNICAL JOURNAL

ample, in a 50-line system the waste bands are each about 500 cycles
wide. In a system using a single sideband of one million cycles width
the waste bands are each about 3300 cycles wide.

These idle frequency regions naturally lead to the questions whether
{a) there is any way of segregating all of the relatively useless signal
components in one region of the frequency spectrum so that the useful
parts of the signal may be transmitted over a channel of about half the
width, or {h) whether it would be worth while placing other communica-
tion channels in these waste regions. It must be realized, however,
that even when the complete frequency space is utilized (by any one of
a number of possible schemes), the required frequency band for trans-
mitting a picture of given detail at a given rate is still only halved as
compared with the simple system considered above, which is not a
change in order of magnitude. The problems of transmitting the wide
band of frequencies necessary, for example, in television, while lessened,
therefore still remain.

APPENDIX I

The calculation of Yi{m, n) according to equation (20') is, for the
three simple apertures here considered, a straightforward mathematical
process which will therefore not be reproduced. The results are plotted
in the form of charts in the conventional manner for functions of two
variables, namely as a series of contours, one of the two variables being
icept constant for each contour. This constant value changes progres-
sively for each successive contour.

The variables are taken as ni and n, multiplied by parameters depend-
ing on the sizes of the scanning aperture and of the picture. Because
of the obvious symmetry of the function, only half of each chart has
been drawn. In order to avoid confusion the contours have been
dotted when \m\ is greater than the first root of Yi{m, 0) = 0. In
one case the contour is shown in a dashed line when \m\ is equal to this
root. Constant factors in the scale of ordinates have been neglected,
to make Fi(0, 0) = 1.

A THEORY OF SCANNING

505

■o

o
E tJ

^t"

<t3

z
in

il
W

w
II

\^*

•i

J^^

w
Ik

'^■^"'^

\ V

<:^

/ d

0J\ ^\

/■

, 1
1 \

1 1

,— I c

EQUIVALENT TRANSFER ADMITTANCE Y(m,n)

506

BELL SYSTEM TECHNICAL JOURNAL

EQUIVALENT TRANSFER ADMITTANCE Y(m,n)

A THEORY OF SCANNING

507

.5f

EQUIVALENT TRANSFER ADMITTANCE Y(nn,n')

508 BELL SYSTEM TECHNICAL JOURNAL

APPENDIX II

As in the case of Appendix I, the calculation of Yi{m, n) • Fo(m', n')
is a straightforward mathematical procedure which will not be repro-
duced. The results are again presented in the form of charts.

In these charts the intensities of the principal extraneous components
are indicated by solid lines, while the higher order extraneous com-
ponents are indicated by dotted lines. The normal components have
been indicated by dashed lines.

It should be explained that what has really been plotted is Fi(m, n)
• Y<i{m, n') rather than Yi{m, n) • F2(m', n'). This is because the
difference between m' and m, when multiplied by the parameters
chosen for the charts, varies with the proportions of the scanning system
used. As discussed in the text with regard to Fig. 17, however, this
difference between m' and m has a negligible effect for any system em-
ploying a useful number of scanning lines.

A THEORY OF SCANNING

509

F'-O.l

-—0.6

- — 0.6

-V''

?■'*'

*^5?ji

■^ V " ^
^X*^^

--,

^::~^

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 06

N N~

^"02^

/ /

^"OA^^

N. \

0.6

/::

0.8_

""•-.

"^v^

,-'

:?-'''

**-^

""-^^sr-i;-:

= = =

:^'

-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 I..

n'=Il 1
N N"

\^=°

h ,

--L-

--M-

—

0.A

SCANNING PITCH =

\o.4
\

0.6

^^v N\

0.8

^\^^

s. ^"'S'- -

:v-- =

-^■=^

"■-■^

..^

y^^-^^^

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2..

Fig. 23 — Rectangular aperture with no overlap.

510

BELL SYSTEM TECHNICAL JOURNAL

> 0

-0.1

0-6 0.8

^0.4

=^""'

«-V,

^^vc-

:;:^

-^ -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

IL'_ Jl_o
N -N "^

0.4

0.2

'(.

0.4

///.

_0.6

''i.o"'

^-,-,

0.8

-;^

::::^

^'"

j:^*'

-1.0 -0.8 -0.6 -0.4 -0.2

0-2 0.4 -0.6 0.8 1.0 I.

n'_n. i

isr"N'^

I.O

0.9

W=o

-N'-N

i'''

0.2\\
\

SCANNING PITCH =
2b 2C
N 1.25

0.4 ^

\
\

"""-

0.6

\ -

i7o~

0.8

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Fig. 24 — Circular aperture with 25 per cent overlap.

A THEORY OF SCANNING

511

\^-

\
\

SCANNING PITCH =

\ \

0.2\ ^
\

0.4 \

\
\

o"^"^

1.0

^^0.8 \

^-~;*i

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Fig. 25 — Diamond shaped aperture with half diagonal overlap.

512

BELL SYSTEM TECHNICAL JOURNAL

-2.0 -1.8 -1.6 -1.4 -1.2

1.0 -0.8 -0.6 -0.4 -0.2

n' _ n _

0 0.2 0.4 0.6

0.2

0
0.1

-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

H' H-i
N ~ N ^

\
\

SCANNING PITCH =

M ^

\
\

0 0.2 0.4 0.6 O.S 1.0 1.2 1.4 1.6 1.8 2.0 Z.Z 2.A 2.6

n
N

Fig. 26 — Sinusoidal aperture with half wavelength overlap.

A THEORY OF SCANNING

513

>. -0.1

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2

n'-n ?

N" N

0.2 04 0.6

^-0.1

-1.0 -0.8 -0.6 -0.4 -0.2 0

0.2 0.4 0.6 0.8 10 1.2 1.4

0.'=I1 I
N N

r'Tf^f-.^^

SCANNING PITCH =

2b _d

N 1-5

■

^^.

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Fig. 27 — Sinusoidal aperture with two-thirds wavelength overlap.

514

BELL SYSTEM TECHNICAL JOURNAL

> 0

^-^

„.^K%Z. I

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

n' n .,

■

«i-

--.-

' /

/ S

.ycA:

E I

-1.0 -0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

JV n
N ~ N

-d^

CASE I

CASE
\ I

SCANNING PITCH :

CASF TT

•

r ~ -

.____

h —

^2b d

SCANNING PITCH = ^ = "g"

02 0.4 0.6 0.6 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

_n_

Fig. 28 — Rectangular aperture with different degrees of overlap.

A THEORY OF SCANNING

515

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

n'_ n -

-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n'_n ,

\
\

< H

SCANNING PITCH =

2b_d

N "2

1
1

\
\

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Fig. 29 — Diamond shaped aperture with three-quarters diagonal overlap.

Abstracts of Technical Articles from Bell System Sources

The Thermionic Work Function and the Slope and Intercept of
Richardson Plots} J. A. Becker and W. H. Brattain. This article
is a critical correlation of the slope and intercept of experimental
Richardson lines with the quantities appearing in theoretical equations
based on thermodynamic and statistical reasoning. The equation
for experimental Richardson lines is log i — 2 log T — log A — b/2.3 T;
A and b are constants characteristic of the surface, i is the electron
emission current in amp./cm.^, T is the temperature in degrees K, log .4
is the intercept and —b/2.3 is the slope of experimental lines. Statis-
tical theory based on the Fermi-Dirac distribution of electron velocities
in the metal shows that i should be given by log i — 2 log T = log
f/(l — r) — w/2.3 T, where C/ is a universal constant which has the
value 120 amp./cm.^ "K^, r is the reflection coefficient, and w is the
work function. A correlation of the experimental and theoretical
equations shows that b = w — Tdw/dt, and log A = log U{\ — r)

— {\l2.3)dwldT. Only when r is 0 and the work function is inde-
pendent of the temperature, is it correct to say that the slope is

— w/2.3 and that the intercept has the universal value log U. But
even when w is a function of T, it follows from a thermodynamic
argument that the slope is given by —h/2.3, where the heat function h
is defined by ^ = {Lp/R) — (5/2) T, Lp is the heat of vaporization per
mol at constant pressure. The heat function is related to the work
function by the equation h = w — TdwjdT.

From experimental and theoretical arguments it is deduced that the
reflection coefficient is probably negligibly small. Hence we conclude
that for most surfaces the work function varies with temperature, since
the intercepts of Richardson lines are rarely equal to log 120. This
conclusion is to be expected since on Sommerfeld's theory, w depends
on the number of free electrons or atoms per cm.\ which in turn varies
with temperature due to thermal expansion.

The photoelectric work function should equal the thermionic work
function but should not in general be equal to —2.3 times the slope of
the Richardson line. The Volta potential between two surfaces having
work functions Wi and Wi should equal {wi — w^ikje rather than 2.3kje
times the difference between the slopes of the Richardson lines for the
two surfaces. The data from photoelectric and Volta potential meas-

^Phys. Rev., May 15, 1934.

516

ABSTRACTS OF TECHNICAL ARTICLES 517

urements support the conclusion that the work function depends on
temperature.

Fundamental Concepts in the Theory of Probability.- Thornton C.
Fry. Three commonly accepted definitions of the word " probability"
are discussed critically, with regard both to logical soundness and to
practical utility. Two major theses are presented: first, that each
definition has utilitarian merits which render it especially valuable
within its own field; second, that the objection of logical redundancy
which is so frequently raised against the Laplacian definition can
equally well be raised against the other two definitions.

Wide-Range Recording.^ F. L. Hopper. The recent improvements
in sound quality resulting from the extension of the frequency and
intensity ranges are the results of coordinated activity in recording
equipment and processes, reproducing equipment, and theater acous-
tics. This paper discusses the recording phase of the process. A wide-
range recording channel consists essentially of the moving-coil micro-
phone, suitable amplifiers, a new recording lens, and certain electrical
networks.

The characteristics of such a system, from the microphone to and
including the processed film, are shown. Other factors fundamentally
associated with wide-range recording, such as monitoring, film proc-
essing, the selection of takes in the review room, and re-recording, are
also discussed. The changes brought about by this system of recording
result, first, in a greater freedom of expression and action on the part
of the actor; and, second, in a much greater degree of naturalness and
fidelity than has been previously achieved.

Iron Shielding for Telephone Cables^ H. R. Moore. Voltages of
fundamental and harmonic frequencies, induced along communication
cables by neighboring power or electric railway systems, can be reduced
by the electromagnetic shielding action of the sheath, if this is grounded
continuously or at the ends of the exposure. The shielding, particu-
larly at the fundamental frequency, is improved greatly by the pro-
vision of a steel tape armor, while a surrounding iron pipe conduit
effects a very great improvement at both the fundamental frequency
and the higher harmonics.

This paper presents methods for the quantitative prediction of
the shielding, expressed by a "shield factor" or the fraction to which

-American Mathematical Monthly, April, 1934.

3 Jour. S. M. P. E., April, 1934.

* Electrical Engineering, February, 1934.

518 BELL SYSTEM TECHNICAL JOURNAL

a disturbing voltage is reduced. Necessary impedance data are given
for numerous iron-surrounded cable constructions and working charts
are supplied for the convenient determination of the shielding obtain-
able with commercially available steel tape armored cables.

On the basis of data presented in this paper, prediction of the
shielding to be obtained from steel tape armored cable sheaths or
those inclosed in iron pipes is concluded to be both feasible and
practical. With internal impedances measurable on short length
samples of a chosen construction, the accuracy of prediction is limited
principally by the precision to which the disturbing field and the
grounding resistances of the cable sheath may be determined. Either
of the constructions discussed is capable of effecting a high order of
shielding against low frequency induction and practically complete
protection from harmonic disturbances. Field observations on in-
stalled cables, both tape armored and in pipe conduit, have verified
the computational methods presented.

Propagation of High- Frequency Currents in Ground Return Circuits.^
W. H. Wise. The electric field parallel to a ground return circuit is
calculated without assuming that the frequency is so low that polariza-
tion currents in the ground may be neglected. It is found that the
polarization currents may be included by replacing the r in Carson's
well-known formulas by r-\jli{e — l)/2cXa-. The problem to be solved
is that of calculating the electric field parallel to an alternating current
flowing in a straight, infinitely long wire placed above and parallel to
a plane homogeneous earth. Carson's derivation of this field is based
on three restricting assumptions: (1) The ground permeability is
unity; (2) the wave is propagated with the velocity of light and without
attenuation; (3) the frequency is so low that polarization currents may
be neglected. The first of these restrictions is usually of no conse-
quence and the formula would be quite complicated if the permeability
were not made unity. As pointed out in a later paper by Carson, the
second restriction amounts merely to assuming reasonably efficient
transmission. The effect of the third restriction begins to be notice-
able at about 60 kilocycles. The object of the present paper is the
removal of the third restriction.

Acoustical Requirements for Wide-Range Reproduction of Sound. ^
S. K. Wolf. The extension of the frequency and volume ranges in
recording and reproducing sound has aroused a greater and more critical

5 Proc. I. R. E., April, 1934.
« Jour. S. M. P. E., April, 1934.

ABSTRACTS OF TECHNICAL ARTICLES 519

consciousness of the importance of theater acoustics. It follows that
higher fidelity in reproduction excites greater intolerance of the needless
distortion caused by poor acoustics of the theater. To cope with the
new situation, engineers have developed new instruments for acoustical
analysis, which provide greater precision and facility in detecting
defects and in determining the necessary corrections.

In addition to instrumental developments there have been concur-
rent advances in acoustical theory and practice. The result is that
the more stringent requirements imposed on the acoustics of the theater
by the enlarged frequency and volume ranges can be fulfilled adequately
and practically. The paper discusses the requirements and describes
some of the available methods for complying with them.

Contributors to this Issue

C. B. Aiken, B.S., Tulane University, 1923; M.S. in Electrical Com-
munication Engineering, Harvard University, 1924; M.A. in Physics,
1925 ; Ph.D., 1933. Geophysical research and exploration with Mason,
Slichter and Hay, Madison, Wisconsin, 1926-28. Bell Telephone
Laboratories, 1928-. Dr. Aiken has been engaged in work on aircraft
communication equipment, broadcast receiver design, centralized radio
systems and common frequency broadcasting.

A. W. Clement, B.S. in Electrical Engineering, University of Wash-
ington, 1925; M.A., Columbia University, 1929. Bell Telephone
Laboratories, Apparatus Development Department, 1925-. Mr.
Clement has been engaged in the development of various types of trans-
mission networks, such as electric wave filters and equalizers.

Karl K. Darrow, B.S., University of Chicago, 1911; University of
Paris, 1911-12; University of Berlin, 1912; Ph.D., University of
Chicago, 1917. Western Electric Company, 1917-25; Bell Telephone
Laboratories, 1925-. Dr. Darrow has been engaged largely in writing
on various fields of physics and the allied sciences.

L E. Fair, B.S., in Electrical Engineering, Iowa State College, 1929.
Bell Telephone Laboratories, Radio Research Department, 1929-.
Mr. Fair has been engaged in experimentation on piezo-electric crystals
for frequency control.

Frank Gray, B.S., Purdue, 1911; Ph.D., University of Wisconsin,
1916. Western Electric Company, Engineering Department, 1919-25.
Bell Telephone Laboratories, 1925-. Dr. Gray has been engaged in
work on electro-optical systems.

H. S. Hamilton, B.S. in Electrical Engineering, Tufts College,
1916. American Telephone and Telegraph Company, Engineering
Department, 1916-18; Department of Development and Research,
1918-34. Bell Telephone Laboratories, 1934-. Mr. Hamilton has
been engaged exclusively in toll transmission work, including telephone
repeaters, program transmission and carrier telephone systems.

F. R. Lack, B.Sc, Harvard University, 1925; Engineering Depart-
ment, Western Electric Company, 1913-22; First Lieutenant, Signal
Corps, A.E.F., 1917-19; Harvard University, 1922-25. Bell Tele-

520

CONTRIBUTORS TO THIS ISSUE 521

phone Laboratories, 1925-. Mr. Lack has been engaged in experi-
mental work connected with radio communication.

W. P. Mason, B.S. in Electrical Engineering, University of Kansas,
1921 ; M.A., Columbia University, 1924; Ph.D., 1928. Bell Telephone
Laboratories, 192 1-. Dr. Mason has been engaged in investigations
on carrier transmission systems and more recently in work on wave
transmission networks, both electrical and mechanical.

R. C. Mathes, B.Sc, University of Minnesota, 1912; E.E., 1913.
Western Electric Company, Engineering Department, 1913-25. Bell
Telephone Laboratories, 1925-. Mr. Mathes has been concerned
with the early history of the repeater development program, the appli-
cation of vacuum tube amplifiers in a variety of fields, and the applica-
tion of voice controlled switching circuits in the toll telephone plant.
As Associate Wire Transmission Research Director he carries on in-
vestigations relating to the transmission of speech over wire systems.

Pierre Mertz, A.B., Cornell University, 1918; Ph.D., 1926.
American Telephone and Telegraph Company, Department of De-
velopment and Research, 1919-23, 1926-34. Bell Telephone Labora-
tories, 1934-. Dr. Mertz has been engaged in special problems in toll
transmission, chiefly in telephotography, television, and cable carrier
systems.

G. W. WiLLARD, B.A., University of Minnesota, 1924; M.A., 1928;
Instructor in Physics, University of Kansas, 1927-28; Student and
Assistant, LTniversity of Chicago, 1928-30. Bell Telephone Labora-
tories, 1930-. Mr. Willard's'work has had to do with special problems
in piezo-electric crystals for frequency control.

S. B. Wright, M.E. in Electrical Engineering, Cornell University,
1919. Engineering Department and Department of Development and
Research, American Telephone and Telegraph Company, 1919-34.
Bell Telephone Laboratories, 1934-. Mr. Wright is engaged in trans-
mission development work on voice-operated systems and wire con-
nections to radio telephone stations.

VOLUME Xra OCTOBER, 1934 number 4

THE BELL SYSTEM

TECHNICAL JOURNAL

DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION

An Extension of the Theory of Three-Electrode Vacuum
Tube Circuits — S. A, Levin and Liss C. Peterson 523

The Electromagnetic Theory of Coaxial Transmission
Lines and Cylindrical Shields — S. A. Schelkunoff 532

Contemporary Advances in Physics, XXVIII — ^The

Nucleus, Third Part— ^arZ K. Darrow .... 580

The Measurement and Reduction of Microphonic

Noise in Vacuum Tubes — D. B. Penick .... 614

Fluctuation Noise in Vacuum Tubes — G. L. Pearson . 634

Systems for Wide-Band Transmission Over Coaxial
Lines — L. Espenschied and M. E, Strieby . . . 654

Regeneration Theory and Experiment — E. Peterson,
J. G. Kreer, and L. A. Ware 680

Abstracts of Technical Papers . 701

Contributors to this Issue 704

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

NEW YORK

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An Extension of the Theory of Three-Electrode
Vacuum Tube Circuits

By S. A. LEVIN and LISS C. PETERSON

The relations between input voltage and output current of the three-
electrode vaccum tube are discussed when arbitrary feedback is present
between grid and plate circuits. Fundamental assumptions are that the
amplification factor is constant and conductive grid current absent. The
relations developed in the present paper are generalizations of those given
by J. R. Carson in I. R. E. Proc. of 1919, page 187. The use of the theory is
illustrated by application to a simple modulator circuit. The numerical cal-
culations in this case indicate that neglecting the effects of interelectrode
tube capacitances may introduce serious'errors.

Introduction

THE relations between input voltage and output current of the
three-electrode vacuum tube when connected to impedances in
both input and output circuits have been the subject of several papers.
One of the first more extensive treatments of this problem was given
by J. R. Carson,^ using a method of successive approximations. The
theory was further extended by F. B. Llewellyn,^ E. Peterson and
H. P. Evans,^ and J. G. Brainerd.* The theories given by these
authors did not take into account any feedback between input and
output circuit except in the first approximation.

The aim of the present paper is to extend the theory of the three-
electrode vacuum tube to include the effects of feedback between
input and output circuits not only in the first but also in the second
and higher approximations. The assumptions underlying Carson's
treatment, constancy of the amplification factor and absence of con-
ductive grid current, will be maintained. The extension of the present
theory to such cases as treated by Llewellyn, Peterson-Evans and
Brainerd still remains to be done.

1 J. R. Carson: I. R. E. Proc, April, 1919, page 187.

2 F. B. Llewellyn: B. S. T. J., July, 1926, page 433.

3 E. Peterson and H. Evans: B. S. T. J., July, 1927, page 442.
< J. G. Brainerd: /. R. E. Proc, June, 1929, page 1006.

523

524

BELL SYSTEM TECHNICAL JOURNAL

Theory

Let us consider the circuit arrangement shown in Fig. 1, where
Zi, Z2 and Zz are linear impedances which may include interelectrode
admittances. The impressed variable electromotive forces whose
instantaneous values are denoted by Eg and Ep are in series with
the impedances Zg and Zp, respectively. In the absence of these
electromotive forces direct currents and voltages are established in
the circuit due to constant grid and plate electromotive forces. With
the variable electromotive forces impressed incremental currents and
voltages are produced. The instantaneous values of these incremental
voltages are indicated on Fig. 1 by g, e, v and p. The incremental
plate current is /. The positive directions of these quantities are
given by the directions of the arrows.

Ep(r\j

Fig. 1 — Three-electrode vacuum tube and circuit.

We will now make two restrictive assumptions: first that the grid
is never positive so that conductive grid current is absent, and second
that the amplification factor ix is constant.

The basis for the analysis is given by the characteristic tube equa-
tion

/ = /(£.+f)

(I)

where I is the total instantaneous current flowing from plate to
filament; Ec is the total instantaneous potential difference between
grid and filament and Eh the total instantaneous potential difference
between plate and filament, ix is the amplification factor. The
relation between the increments e, v and / is given by the following
equation :

J = Piifxe + v) + P^ifxe + z;)2 + • • • + Pnifie + v)- +

(2)

THEORY OF THREE-ELECTRODE VACUUM TUBE CIRCUITS 525

where

_ 1 d-I

"* 7n ! dE^^

and has to be evaluated at the operating point.^
We have further:

E,^ g + e, Ep = p + v. (3)

The equations (3) are obtained by applying the circuital laws to the
network external to the tube.

We now proceed to a solution of equations (2) and (3) by means of
a method of successive approximations. Let

J = II Ji, g= ILgu e = X, eu (4)

111

00 00

P = HPu V = ^v.

and let us define the relations between the terms in the series (4) as
follows :

Ji = Piif^ei + i^i), Eg = gi + ei, Ep = pi + Vi, (5)

J2 = Px{ne2 + V2) + P2(//ei + z;i)2, (6)

0 = g2 + 62, 0 = /?2 + V2,

Jz = PiC^es + ^3) + 2i'2(M^i + z^i)(m^2 + v^ + P3(/xei + ^'l)^ (7)

0 = g3 + ^3, 0 = ^3 + Vz,

Ji = Piiixei + Vi) + P2{tie2 + ^2)^ + 2P2{fj.ei + t'OC^^s + Vz)

+ 3PzifJie2 + V2){uie, + v,y + P4()uei + ^;l)^ (8)

0 = g4 + ^4, 0 = ^4 + ^4,

and so forth for subsequent terms. ^
If we now let

^0 = ^^, (9)

' Loc. cit.

^ The procedure of finding these equations is as follows: By substituting the first
term in each of the series (4j into (2) and (3j and neglecting all terms higher than
the first order equations (5) are obtained. By substituting the first two terms in
each of the series (4) into (2) and (3), and neglecting terms of higher order than the
second and by noting (5) equations (6) are found and so on for the remaining equa-
tions.

526

BELL SYSTEM TECHNICAL JOURNAL

where Rq is the internal resistance of the tube, equations (5), (6), (7)
and (8) may be rewritten as :

RoJi — vi = nei, Eg = gi + ei, Ep = pi + vi,
R0J2 - vi = ixei + RoPiiixei + ViY,

0 = g2 + 62, 0 = ^2 + V2,

(10)

(11)

RJz -vz = nez + 2RoP2(fJiei + Vi)(fjLe2 + V2) + R^Pzinei + v^Y (12)
Q = gz + ez, Q = pz + Vz,

R0J4 — Vi = ixei + RoPiiixe^ + v^Y + IRaPiiiiei + z;i)(/ie3 + Vz)

+ 3RoPz{uie2 + V2) {ixei + v^f + RoP^ifJiei + Vi)', (13)
0 = g4 + ei, 0 = pi -{- Vi,

and so forth.

Equations (10) to (13) admit of simple physical interpretations.
Referring first to equations (10) it is clear that the equivalent circuit
corresponding to Fig. 1 for first order quantities is given by Fig. 2.
Similarly Fig. 3 is the equivalent circuit of Fig. 1 for second order
effects and Fig. 4 for third order effects. Higher order effects corre-
spond to similar circuits.

JiJ|ro

Fig. 2 — Equivalent circuit, first order effects.

The equivalence expressed by Fig. 2 is the familiar circuit which
has found such wide application, for instance, in amplifier and oscil-
lator work; while the equivalent circuits in Figs. 3 and 4 represent
the second and third order effects. With no feedback, that is when
Z2 is infinite, they reduce to the equivalences given by Carson.^ Com-
paring now any two equivalent circuits for same order effects with and
without feedback we find different values of the electromotive forces
appearing in series with the internal tube resistance Rq. Otherwise

1 Loc. cit., equations (23) and following.

THEORY OF THREE-ELECTRODE VACUUM TUBE CIRCUITS 527

the two circuits are identical except that for one the impedance Z2 is
finite and for the other infinite.

By the aid of the equivalent circuits given, that is by using equations
(10), (11), (12), (13) and so forth, the terms in the series (4) can be
calculated. These series formally satisfy equations (2) and (3) and
are the solutions if they converge.

Fig. 3 — Equivalent circuit, second order effects.

2R0P2 O^e, + Vi)(jjLe2 + V2)+ [(^
RoP3(>xe,^vy

Fig. 4 — Equivalent circuit, third order effects.

For the purpose of fixing our ideas we assumed at the start a definite
circuit to which the tube was connected. It is obvious, however,
that no matter how complicated the linear network is to which the
input and output terminals of the tube are connected the procedure
given above can be followed.

Application to a Modulator Circuit

As an illustration of the theory just presented we shall calculate
the steady state second order effect assuming the circuit configuration
to be that given in Fig. 1. In so doing we shall assume that no variable

528

BELL SYSTEM TECHNICAL JOURNAL

e.m.f. is impressed in the plate circuit and that the impressed e.m.f.
in the grid circuit is given by ^

K cos Oilit -\- S cos C02^.

(14)

We now find the instantaneous value of tiCi -\- Vi by solving the mesh
equations for the equivalent circuit of Fig. 2, The result is:

ixei + Vi, = Ro

where

F{o:) =

K cos (coi/ — ^(«i))
F{w^

6* cos {(Jilt — (p{u2))

Z(C02)

Zi(juZ2 + iJ'Zp + Z/)
(Z, + Zi)(Z/ + Z2) '

Z/(i?o + fiZ/)

(15)

Z(co) = i?o + Z/ +
ZsZp

7 ' =

FM =
Zic)

Zz + Zp

7 ' =

Z2 + Z/

Z\Zn

(16)

Zi + Z„

Z(co)

g-v(")i(i = V- 1).

In equations (16) we note that Zi, Z2, Z3, Z^ and Zp all are complex
impedances. The driving e.m.f. for the second approximation is

RoPiiliei -h v^f. Letting

M = Ro'P2,

we get from (15)
RoP2(.tJ^ei + ViY
= M

(17)

Z(coi)

i^2 +

i^(w2)

Z(C02)

+
+
+
+

Z(coi)
^(0)2)

Z(C02)

F(o}i)F(o)2)

Z(coi)Z(aj2)

7^(cOl)F(c02)

Z(coi)Z(aj2)

X2 COS (2co,t - 2<p(coi))
S^ cos (2co2^ — 2^(0)2))

^"6" COS ((cOi — CO2)/ — ^(cOi) + <P(^2))

(18)

i^5 COS ((coi + 002)^ — v'(wi) ~ ^(''^2))

^ The extension to any number of sinusoidal e.m.f.'s of arbitrary phases in both
plate and grid circuit is obvious.

THEORY OF THREE-ELECTRODE VACUUM TUBE CIRCUITS 529

The driving e.m.f. given by (18) thus consists of a number of
sinusoidal components including one of zero frequency. By means
of the superposition theorem and the mesh equations we obtain the
current and voltage distribution for our equivalent circuit in Fig. 3.
Let us for instance calculate the instantaneous current flowing through
the impedance consisting of Zp and Zz in parallel and indicated by C in
Fig. 3. The result is

C= M

F(coi)
Z(coi)

K^ +

F(C02)

Z{wi)

Z(0)

+ •

F(a:{)

ZiccO

\Z(2co,)
FM

•COS (2coi^ — 2<p(o}i) — \p{2coi))

Zicoi)

-f

|Z(2C02)|
F(cOi)F(c02)

•COS (2co2^ — 2^(aj2) — ^(2^2))

(19)

Z(cOi)Z(c02)

|Z(coi — 0)2)

F(0}i)F(c02)

COS ((oji — coo)^ — ip(coi)

+ ^(0)2) — i/'(wi — CO2))

Z(coi)Z(a;2)

|Z(coi + CO2)

COS ((wi + 0}2)t — (p{o}i)

— <p{(^2) — ^(oOl + W2))

where \p(co) is defined by

Z(co) = |Z(co)!e^^(")(^ = V^T).

(20)

Let us now consider the peak value of the current of lower side-band
frequency. This value is from (19)

MKS

F(i,}i)F{ojo)

If we write

|Z(co)|

i?fl

Z(coi)Z(aj2)Z(coi — 0:2)

1 +

Ro + fxZ,

- A- 7 '

Z2 + Z/

F(co)|= (m+1)

1 +

Z2 + ZJ

(z, + Zi)(z; + Z2)

M + 1

Z2 + Zp

(21)
(22)

(23)

530 BELL SYSTEM TECHNICAL JOURNAL

expression (21) becomes:

MKSitx + \y

(Z, + Zi)(Z/ + Z2)

IJ-

M+ 1

Z2 4" Zp

M+ 1

Z2 + Zp

1 -F

i?o + mZ/
Z2 + ZJ

Z/+-

1 +

Ra + AtZff'

i?o + mZ/
Z2 + Z/

Z/ +

1 +

i?o + mZ/

1 +

Z2 + z/

Zo + ZJ

(24)

(Wj— w,)

1 +

Ro + A1Z9'

Z/ +

Z2 + Z/
Ro

1 +

Z2 + Z/

("1-6)2)

With no feedback present, that is when Z2 = <» , equation (24)
reduces to

MKSfx^

0)1

Zi 1

Zr + Z,

Z: + zJ

But in this case K

Zp + -^0 1 0), I Zp' + i^o UJ Zp' + Ro I (wi-coj)
Zi

(25)

Zx + Z
of frequency coi and similarly S

is the peak value K' of the grid voltage
Zi

is the peak value S' of the

Zl + Z2 I 0)2

grid voltage of frequency C02. Expression (25) may thus be written :

I Zp 4 -/?0 I 0)J Zp' + -^0 I 0)2 I Zp + i?o I (0)1-0.2) '

which is the well known expression given by Carson.'^

For the purpose of getting an idea of the magnitudes involved let
us consider a numerical example. A Western Electric No. 101 D
vacuum tube may have the following constants when used as a modu-
lator: Ro = 9000 ohms; ix = 6; grid-cathode capacitance Ci = 10.5,
plate-grid capacitance d = 4.8, and plate-cathode capacitance C3
= 8.1 micromicrofarads. The impedances Zp and Zg are assumed to
be pure resistances at all frequencies with the values 9000 and 10,000
ohms, respectivel3^ The impressed e.m.f. is of the form given by

' Carson: I. R. E. Proc, June, 1921, page 243.

THEORY OF THREE-ELECTRODE VACUUM TUBE CIRCUITS 531

the equation (14) where coi/27r is equal to 250,000 and cu2/27r is equal
to 5,000,000 cycles per second. As a reference condition let us take
that for which the effect of the interelectrode capacitances is neg-
lected. The plate current for this condition is obtained from equation
(25) when Zi and Zz as well as Z2 are made infinite. As a next step
we compute the plate current from equation (24) when Zi, Z2, and Z3
are the impedances corresponding to the interelectrode capacitances
C\, C2, and C3, respectively. It is found that this plate current is
12 db below that obtained in the reference condition. Finally it is
of some interest to compute the plate current when the grid-plate
capacitance alone is effective. This plate current is obtained from
equation (24) by assuming Zi and Z3 to be infinite and Z2 to be the
impedance corresponding to the capacitance d. This current is
found to be 24 db below that of the reference condition.

The Electromagnetic Theory of Coaxial Transmission Lines
and Cylindrical Shields

By S. A. SCHELKUNOFF

A form of circuit which is of considerable interest for the transmission
of high frequency currents is one consisting of a cylindrical conducting
tube within which a smaller conductor is coaxially placed. Such tubes
have found application in radio stations to connect transmitting and re-
ceiving apparatus to antennae. As a part of the development work on such
coaxial systems, it has been necessary to formulate the theory of trans-
mission over a coaxial circuit and of the shielding against inductive effects
which is afforded by the outer conductor. This paper deals generally
with the transmission theory of coaxial circuits and extends the theory
beyond the range of present application both as regards structure and
frequency.

THE mathematical theory of wave propagation along a conductor
with an external coaxial return is very old, going back to the
work of Rayleigh, Heaviside and J. J. Thomson. Much important
work has been done in developing and extending this theory. Among
the problems dealt with in this development may be listed the follow-
ing: the extension of the theory to systems consisting of a plurality
of cylindrical conductors; the investigation of shielding and crosstalk
in coaxial systems and the effects of eccentricity; the extension of the
particular solution to include the complementary modes of propaga-
tion, etc.; and in general the adaptation of the mathematical theory
to engineering uses, and its translation into the concepts and language
of electric circuit theory. In addition to the author's contribution a
substantial part of this mathematical work has been done by the
group of engineers associated with Mr. John R. Carson, formerly of
the American Telephone and Telegraph Company, now of the Bell
Telephone Laboratories, Inc.

The problem is ideally adapted to mathematical investigation,
because the conductor shape fits perfectly into the cylindrical system
of coordinates, thereby making it entirely feasible to carry out a
rigorous discussion on the basis of the electromagnetic theory, instead
of using ordinary circuit theory. This has obvious advantages at
ultra high frequencies, where the uncertainties of the circuit theory are
conspicuous and not easily compensated for. It also proves to be
of greater advantage at lower frequencies than one might at first
assume. Fortunately, it turns out that the final results obtained by
means of field theory can be expressed in a familiar language of circuit

532

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 533

theory, thereby gaining all the simplicity of the latter combined with
all the accuracy of the former.

Circularly Symmetric Electromagnetic Fields
In polar coordinates, Maxwell's equations assume the following
form :

7a^ - ^ = (« + "")^"
^' - ^' = fe + i-)£.,

dE,

dp dip

djpE^) _ dEp

dp d(p

= (g + io:e)E^,

= — ioip-H^,

(1)

where E and H are respectively the electromotive and magnetomotive
intensities.'

In general, all six field components depend upon each other. If,
however, these quantities are independent of either tp or z, the partial
derivatives with respect to the corresponding variable vanish and the
original set of equations breaks up into two independent subsets, each
involving only three physical quantities. Each of these special fields
has important practical applications.

In the circularly symmetric case, that is, when the quantities are
independent of ^, one of the independent subsets is composed of the
first and the third equations on the left of (1), together with the second
on the right:

d{pH^)
dp

(g + icc€)pE„

dE. dEn

dH,
dz

= - (g + io:e)Ep,

(2)

= ioijxH^.

This circular magnetic field, with its lines of magnetomotive intensity

^ In this paper we have adopted a unified practical system of units based upon
the customary cgs system augmented by adding one typically electric unit. This
system has three obvious advantages: first, theoretical results are expressed directly
in the units habitually employed in the laboratory; second, the dimensional character
and physical significance of such quantities as io>ix and g + io^e are not obscured as
in other systems by suppressing dimensions of some electrical unit such as permea-
bility or dielectric constant; and third, the form of electromagnetic equations is very
simple. In this system of units the electromotive intensity E is measured in
volts/cm., the magnetomotive intensity H in amperes/cm., the intrinsic conductance
g in mhos/cm., the intrinsic inductance ix in henries/cm., and the intrinsic capacity e
in farads/cm. Thus, in empty space /x = 47rl0~^ henries/cm. or approximately
0.01257 nh/cm. and e = (l/367r)-10~" farads/cm, or approximately 0.0884 mmf./cm.

534

BELL SYSTEM TECHNICAL JOURNAL

forming a system of coaxial circles, is associated with currents flowing
in isolated wires as, for example, in a single vertical antenna and under
ordinary operating conditions it is also found between the conductors
of a coaxial pair (Fig. 1).

Fig. 1 — The relative directions of the field components in a coaxial transmission line.
The remaining three equations of the set (1) form the second group:

a(p£,)

= — iwfxpHz,

(g + iue)E^ =

dz
dH, dH

- = ioofxHp,

(3)

dz dp

describing the circular electric field. Uniformly distributed electric

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 535

current in a circular turn of wire is surrounded by a field of this type;
in this case, the lines of electromotive intensity form a coaxial system
of circles.

TWO-DIMENSIONAL FlELDS

By definition, two-dimensional fields are constant in some one
direction. If we take the z-axis of our reference system in this direc-
tion, all the partial derivatives with respect to z vanish, 2 disappears
from our equations and we can confine our attention to any plane
normal to the z-axis.

Once more the set of six electromagnetic equations breaks up into
two independent subsets. One of these is -

1 dH, 1 dH.

(g -f- icoe)p dip ' g + -icoe dp

d(pE^) ^ dE,

(4)

= — ioinHf

dp d(p

The calculation of what is commonly known as "electrostatic" cross-
talk between pairs of parallel wires is based upon these equations.
For this reason we shall name the field defined by (4) the electric
field.

Similarly, the remaining three equations define the magnetic field :

. , _ _ J_ aE, zr - _ X ^^

loifxp ocp iwp. op

dp dtp

(5)

{g + iwe)E^

and are useful in the theory of what is generally known as "electro-
magnetic" crosstalk.

The distinction between electric and magnetic fields is purely prag-
matic and is based upon a necessary and valid engineering separation
of general electromagnetic interference into two component parts.
In some respects the firmly entrenched terms "electrostatic crosstalk"
and "electromagnetic crosstalk" are unfortunate; it would be hopeless,
however, to try a change of terminology at this late stage of engineering
development.

Further consideration of two-dimensional fields will be deferred
until the problem of shielding is taken up later in this paper (page
567).

2 In passing from the original set (1) we reversed the sign of Ep in order to make
the set of equations symmetrical. The positive Ep is now measured toward the axis.

^ In these equations, the sign of H^ was reversed so that the magnetomotive
intensity is now positive when it points clockwise. With this convention, the
flow of energy is away from the axis when both H^ and Et are positive.

536 BELL SYSTEM TECHNICAL JOURNAL

Exponential Propagation

While electromotive forces could be applied in such a way that the
fields would be of the kind given by (3), in the coaxial transmission
line as actually energized the fields are of the circular magnetic type
(2) which will claim our special attention in the next few sections.

In order to solve equations (2), we naturally want to eliminate all
variables but one. This purpose can be readily accomplished if Ej
and Ep are substituted from the first and the last equations of the set
into the second. Thus, we obtain the following equation for the
magnetomotive intensity:

+ -^= C.W,, (6)

d_ ri djpH^y

dp Lp dp
where

a^ = gwiii — aj^€ju. (7)

Adopting the usual method of searching for particular solutions of
(6) in the form

H, = R{p)Z{z), (8)

where R{p) is a function of p alone, and Z{z) a function of s alone,
we get

l^=p (9)

1 ^j ^ ^, _ ^,^
P dp J

1 A
Rdp

where V is some constant about which we have no information for
the time being.

Equation (9) is well known in transmission line theory; its general
solution can be written in the form

Z = Ae^' + Be-^', (11)

where A and B are arbitrary constants. The solutions of (10) are
Bessel functions. Since equation (6) is linear, we may invoke the
principle of superposition and add any number of particular solutions
corresponding to different values of F. Thus we can form an infinite
variety of other solutions so as to satisfy the physical conditions of
various practical problems.

It is seen at once from the first and the last equations of the set (2)
that to each H^ of the form (8) there correspond an Ez and Ep of the
same form; i.e., there exist circularly symmetric electromagnetic

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 537

fields, all of whose components vary exponentially in the direction of
the axis of symmetry. Whether any of these fields can be produced
individually by some simple physical means is impossible to decide on
theoretical grounds alone. It may happen, of course, that the field
due to any practically realizable source is always a combination of
several simple exponential fields. In any case, however, we want to
know the properties of pure exponential solutions.

It is convenient to make the exponential character of the quantities
Ep, Ez and H^ explicit and write them respectively in the form Epe~^^,
Eze~^^ and H^e~^^. The new quantities Ep, Ez and H^ are functions
of p only. If the suggested substitution, is made in equations (2),
the factor e~^^ cancels out and we have

r dE

!^ = (g + .-cOpB..

The quantity T is called the longitudinal propagation constant or
simply the propagation constant when no confusion is possible.*

Recalling the implied exponential time factor e^"', we see that the
complete exponential factor in the expressions for the field intensities
is e~^^+^"^ The propagation constant T is often a complex number
and can be represented in the form a + i^ where the real part is
called the attenuation constant and the imaginary part, the phase
constant. Thus, e~"^ measures the decrease in the amplitudes of the
intensities and g-»(^2-w'), the change of their phases in time as well as
in the z-direction. The latter factor suggests that we are dealing
with a wave moving in the positive direction of the 2-axis with a velocity
(co/jS). A wave moving in the opposite direction is obtained by re-
versing the sign of F.

Perfectly Conducting Coaxial Cylinders ^
Let us now consider one of the simplest problems which, though
purely academic in itself, will throw some light on what is likely to
happen under less ideal conditions. We suppose that a perfect
dielectric is enclosed between two perfectly conducting coaxial cylinders
(Fig. 1) whose radii ^ are b and a {b < a). Our problem is to. find the
symmetric electromagnetic fields which can exist in such a medium.

* Another set of exponential solutions is obtained from this by changing r into — T.

* For a thorough discussion of "complementary" waves in coaxial pairs the reader
is referred to John R. Carson [4].

^ Only the outer radius of the inner conductor and the inner radius of the outer
conductor need be considered because in perfectly conducting media electric states
are entirely surface phenomena.

538 BELL SYSTEM TECHNICAL JOURNAL

In a perfect dielectric g = 0 and the preceding set of equations becomes

r dE

Ep = : — H^, iunH^ = -^ + VEp,

tcoe dp , ^

No force is required to sustain electric current in perfect conductors
and the tangential components of the intensities are continuous across
the boundaries between different media; therefore, the longitudinal
electromotive intensity vanishes where p equals either a or h.

Substituting Ep from the first equation into the second, solving
the latter for H^ and inserting it into the third equation, we have
successively

H,^-i^^, (14)

and

m^ dp

P^ + ^ + w^pE. = 0, (15)

where, for convenience, we let F^ + w^e// = in^. The most general
solution of the last equation is usually written in the form

£.(p) = AJo(mp) + BYo(mp), (16)

where Jo and Yo are Bessel functions of order zero and A and B are
constants so far unknown^

The constants A and B can be determined from the fact already
mentioned that Eg vanishes on the surface of either conductor, i.e.,
from the following equations :

AJoimb) + BYoimb) = 0,

and (17)

AJo{ma) + BYoima) = 0.

These equations are certainly satisfied if both constants are equal
to 0. If, however, they are not equal to 0 simultaneously, we can
determine their ratio from each equation of the above system. These
ratios should be the same, of course, and yet they cannot be equal for
every value of m. Thus, the permissible values of m are the roots of

^ For large values of the argument these Bessel functions are very much like
slightly damped sinusoidal functions; in fact Joix) and Yoix) are approximately
equal, respectively, to yll/irxcos {x — w/i) and yjl/irx s\n (x — 7r/4), provided x is
large enough.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 539
the following equation:

A Yo(mb) Y(,(ma)

B Jo{mb) Joima)

(18)

This equation has an infinite number of roots * whose approximate
values can be readily determined if we replace Bessel functions by
their approximations in terms of circular functions. Thus, we have

«7n = -^, (w = 1,2, 3, •••). (19)

This is a surprisingly good approximation for all roots if the radius of
the outer conductor is less than three times that of the inner; and the
larger the n, the better the approximation.^ The propagation con-
stants are computed from the corresponding values of w„ by means
of the following equation,

r„ = \m„2 - co^e/x. (20)

First of all, let us study the simplest solution in which both A and
B vanish. In this case, the longitudinal electromotive intensity
vanishes identically. The magnetomotive intensity — and the trans-
verse electromotive intensity, as well — also vanishes unless the de-
nominator m^ in equation (14) equals zero. If all intensities were to
vanish, we should have no field and there would be nothing to talk
about; hence, we take the other alternative and let

r2 + cahti = 0, i.e., r = icosliT, (21)

the positive sign having been implied in writing equations (13). In
air, e/i = (l/c^) where c is the velocity of light in cm.; hence, in air
this particular propagation constant equals iu/c. Since E^ equals
zero everywhere, the electromotive intensity is wholly transverse; and
the flow of energy being, according to Poynting, at right angles to the
electromotive and magnetomotive intensities, the energy transfer is
wholly longitudinal.

The above method of determining the propagation constant may
be open to suspicion ; besides, the method does not tell how to obtain
the actual values of the electromagnetic intensities but merely leads
to a relation compatible with the existence of such intensities. There-
fore, let us obtain the wanted information directly from the funda-

* A. Gray and G. B. Mathews, "A Treatise on Bessel Functions" (1922), p. 261.
^ It is strictly accurate if the radii of the cylinders are infinite, i.e., if we are dealing
with a dielectric slab bounded by perfectly conducting planes.

540 BELL SYSTEM TECHNICAL JOURNAL

mental equations (13) which assume the following simple form:

ii^eE, = TH^, ii^ull^ = TE„ "EeEA = 0, (22)

if Eg vanishes identically. Either of the first two equations determines
the ratio of the electromotive intensity to the magnetomotive; the
two ratios are consistent only if the condition (21) is satisfied. Then,
we have also

r _ t .. , ^^ A

Ep =-^ H^ = ^p 11^ and H^ = - , (23)

tcoe \ € p

where A is some quantity independent of p. This constant can be
readily calculated from Ampere's law. The magnetomotive force
acting along the circumference of any particular cross-section of the
inner cylinder equals l-wpH^ amperes, i.e., lirA; since this M.M.F.
should equal the total current I flowing in the inner conductor through
the cross-section, the quantity A equals 7/2 tt. Reintroducing the
implied factor e"~^^, we have

■'■■'■ V rj ^ )

/xp

27rp \ e

In practical measurements we are concerned with the total potential
difference (F) between the cylinders, rather than with the transverse
electromotive intensity. The former is merely the integral of the
intensity.

This voltage and the current I vary as voltage and current in a semi-
infinite transmission line whose propagation constant is T and whose
characteristic impedance is

At any point z the intensities Ep and H^ have the same values as ivould
the voltage and current at the same distance z from the end of a trans-
mission line whose propagation constant and characteristic impedance are
respectively icoVcAi and ^ult.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 541

The connection between electromagnetic theory and line theory is
so important that, risking repetition, we wish to emphasize their
intimate relationship by deriving the well-known differential equations
of the line theory directly from the electromagnetic equations (2)
combined with the assumption that the longitudinal electromotive intensity
vanishes everywhere. We already know that under the assumed con-
ditions the first equation of the system (2) becomes

where / is the total current flowing in the inner cylinder through a
particular cross-section and is some function ^" of s. We can therefore
rewrite the last two equations of the system as follows :

dEp icofj. ^ I dl .

oZ LTrp Airp oz

We have merely to integrate both equations with respect to p from
b to a and substitute the potential difference V for the integral of the
transverse electromotive intensity to obtain

dV / icofx a\ dl Iwicoe

a^= -Ur^^^^j^' Tz=-T-a:^' (29)

log^

which are the equations of the transmission line whose distributed
series inductance equals (fxllir) log (a/b) henries/cm. and shunt capacity
27re/(log a/b) farads/cm.

With this, we conclude the special case in which the longitudinal
electromotive intensity vanishes everywhere, the propagation constant
equals icoVe^t, and the velocity of transmission is that of light.

We now turn our attention to the case in which A and B do not
vanish. We have already noted that the propagation constants are
given by equation (20). Since, in this case, we are interested primarily
in the nature of the phenomena rather than in the details of field
distribution, we shall simplify our mathematics by supposing the
radii of the cylinders to be infinite. Thus, the cylinders become two
planes perpendicular to the x-axis, distance a apart. The 99-direction,
then, coincides with the 3;-direction and, therefore, all the intensities
are independent of the 3;-coordinate. Let us choose the z-axis half-
way between the planes. The equations describing this two-dimen-

" On this occasion, we should remember that a particular type of this function
had not yet been ascertained at the time the equations (2) were arrived at.

542 BELL SYSTEM TECHNICAL JOURNAL

sional transmission line are

dlly . „
dx

-—^ = - iweE:,, (30)

dEx _ dEz _ _ . jT

dz dx ^'

If n is an odd integer, these possess the following solutions:

. r„ . nvx

Ex = A - — sin ,

iwe a

Ez = A-. cos , (31)

tojea a

Hy = A sin ;

and if n is an even integer,

r„ n-rrx

Ex = A -. — cos ■ ,

twe a

E,= —A-. sin , (32)

tcjea a

mrx

Hy = A COS

where

and X is the wave-length corresponding to the frequency /.

Let us now define the longitudinal impedance (Z^) as the ratio of

Ex to Hy,

Z.=^, (34)

and the transverse impedance (the impedance in the x-direction) as
the ratio of E^ to Hy,

^ nir mrx .. . ,,

Zx = -■ cot , 11 w is odd,

^o:ea a ^^^^

„ mr mrx ..

Zi = — -: — - tan- ■ , 11 w IS even.

twea a

It will be observed that, depending on the frequency, the longitudinal

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 543

propagation constant r„ is either real or purely imaginary; it vanishes
if o = w(X/2), that is, if the spacing between the planes is a whole
number of half wave-lengths. When the propagation constant is
real, the longitudinal impedance is purely imaginary, and vice versa,
when the propagation constant is purely imaginary, the longitudinal
impedance is real. In the former case, no energy is transmitted
longitudinally but merely surges back and forth, and in the latter
case we have a true transmission line. The transverse impedance is
purely imaginary at all frequencies and, hence, the energy merely
fluctuates to and fro.

If the frequency is sufficiently low, all of these higher order propaga-
tion constants are real and all the energy is transmitted in the principal
mode described by equations (21) to (29). The role of the higher
propagation constants consists in redistributing the energy near the
sending terminal, ^^ that is, in terminal distortion. But as the fre-
quency gets high enough to make the wave-length less than 2a, the
next transmission mode may become prominent, and so forth up the
infinite ladder of transmission modes.

Imperfect Coaxial Conductors ^^
We shall now suppose that the conductors are not perfect; i.e.,
the conductivity instead of being infinite, is merely large. Assuming
that our solutions are continuous functions of conductivity (this can
be proved), we conclude: first, there exists an infinite series of propaga-
tion constants approaching the values given in the preceding section
as the conductivity tends to infinity; second, one of these propagation
constants, namely that approaching ^'coVe^t, is very small unless the
conductivity is too small. In the immediately succeeding sections we
shall be concerned only with electromagnetic fields corresponding to
this particular propagation constant.

Let us now prove that the simple expression for the magnetomotive
intensity in the dielectric between perfectly conducting cylinders is
still true for all practical purposes, even if the conductors are merely
good, and even when there are more than two of them. Since the
lines of force are circles, coaxial with the conductors, and since H^ is
independent of <p, the total magnetomotive force acting along any
one of the circles equals H^ times the circumference of the circle {2irp).
This M.M.F. also equals the total current / passing through the area
of the circle. Therefore, the magnetomotive intensity is (7/2 xp)
amperes/cm. This expression is true at any point in the conductors as

" And near the receiving terminal as well, if the line is finite.
1* The general theory of wave propagation in a multiple system of imperfect
coaxial conductors is amply covered by John R. Carson and J. J. Gilbert [2, 3].

544 BELL SYSTEM TECHNICAL JOURNAL

well as in the dielectric between them. In a conductor the total
current / passing through the area of the circle is a function of p since
the current is distributed throughout the entire cross-section of the
conductor. Strictly speaking, the same is true of any circle in the di-
electric. There is one important difference, however; the conduction
current passing through such circles is the same and the displacement
current is usually so small that it can be legitimately neglected.
Thus, in the dielectric, we have to an extremely high degree of accuracy
unless p is very large

H,=^, . (36)

Z7rp

where / is merely a constant, namely, the total conduction current
passing through the area of the circle of radius p.

That the longitudinal displacement current can be neglected, unless
the conductivity of the conductors is small, has been already indicated
in the opening paragraph. The following comparison is an aid to the
mathematical argument. The density of the longitudinal conduction
current is gE and that of the displacement current is ioieE. Near the
boundary, E is substantially the same iil the conductor and in the di-
electric. In copper, g = (1/1.724)10« and in air e = (l/367r)10-ii.
Thus, even at very high frequencies, the density of the displacement
current is very small compared to that of the conduction current.
On the other hand, the conduction current is ordinarily distributed
over a small area while the displacement current may flow across a
large area. The latter area would have to be very large, however,
before it could even begin to compensate for the extremely low current
density.

Electromotive Intensities in Dielectrics

With the aid of equations (12) and (36), we can now calculate the
electromotive intensities in the dielectric between two conductors.
Thus, the transverse intensity is

Substituting this in the second equation of the set (12), we obtain
the following differential equation for the longitudinal intensity:

dE,
dp

ICjOfJ, ~

2^' (^^)

Zirp

g + ioje
where m is the permeability of the dielectric. Integrating with respect

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 545
to p, we have

E =J-

/log^ + ^, (39)

g + ^coe
where ^ is a constant to be determined from the boundary conditions.^^

The Potential Difference Between Two Coaxial Cylinders

Equation (36) relates the transverse electromotive intensity to the
total current flowing in the inner conductor. In practice, however,
we are interested in the difference of potential between the conductors,
that is, in the transverse electromotive force rather than the electro-
motive intensity. This potential difference V is obtained at once
from equation (37) by integration:

V = f E,dp = ^ ,^! . , f

d, r/log«^

p 2x(g + icoe)

(40)

This transverse E.M.F. produces a transverse electric current which
is partly a conduction current — if the dielectric is not quite perfect — •
and partly a displacement (or "capacity") current.

Now, the total transverse current per centimeter length of line is

Ip = 2irp{g + iuie)Ep.

Then, by equation (37), we have

Ip = TL (41)

Therefore, equation (40) becomes

logy

27r(g + ijcoe)

The ratio of a current to the electromotive force that produces it
is called admittance. Hence, the distributed radial admittance per

1' The following system of notation will be adhered to throughout the remainder
of the paper: The inner radius of any cylindrical conductor is denoted by a, and its
outer radius by b. When several coaxial conductors are used, they are differen-
tiated by superscripts; a' , a" , a*^*, • • • referring to their inner radii, for example,
and b' , b", i<^', • ■ • to their outer radii. This convention also applies to conduc-
tivities, permeabilities, and other physical constants of the conductors in question.

For convenience, we have written the ratio of p to the outer radius of the inner
conductor in place of p; this change affects only the arbitrary constant A which will
eventually be assigned the value required by the boundary conditions. When
written in this form, the first term of E^ vanishes on the surface of the inner con-
ductor which is a convenience in determining the value of A.

546 BELL SYSTEM TECHNICAL JOURNAL

unit length between two cylindrical conductors is

Y = ^"fe +Jr^ ^ G + i^C, (43)

logy

the symbols G and C being used in the usual way to designate the
distributed radial conductance and capacity. Writing these sepa-
rately, we have

r> ^TTg 27re . .

G = ^ , C = ^, . (44)

logy log-y

Returning to (40), we find that F can be written in the form

But the ratio of the transverse electromotive force V to the longitudinal
current / is known as the longitudinal characteristic impedance of the
coaxial pair. Its value is obviously F/F.

The External Inductance

In dealing with parallel wires it is customary to use the term
"external inductance" for the total magnetic flux in the space sur-
rounding the pair.^^ We shall adopt the same usage in connection
with coaxial pairs. Strictly speaking, we must therefore consider it
as being composed of two parts : one being the flux between the cylinders,
the other the flux in the space surrounding them. But the longi-
tudinal displacement current is negligible by comparison with the con-
duction current, and effects due to it have been consistently ignored
throughout this part of our study. To the same order of approxima-
tion, the flux outside the pair is negligible by comparison with that
between them, whence we find the "external inductance" to be

M j ^ H^dp ^„

Le = ^' J = TT- log ^T henries/cm. (45)

1 ATT 0

" While this definition is very descriptive, it is not strictly accurate unless the
wires are perfectly conducting. The correct definition should read as follows:
The external inductance of a parallel pair is the measure (per unit current) of mag-
netic energy stored in the space surrounding the pair. The reason the simpler
definition fails for imperfectly conducting parallel wires is because some of the lines
of magnetic flux lie partly inside and partly outside the wires. This does not happen
in connection with coaxial pairs even when they are not perfectly conducting.
Hence we are warranted in using the simpler idea.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 547

Comparing this with equation (44), we have the following relation
between the external inductance and the capacity

CLe = etx. (46)

Propagation Constants of Coaxial Pairs

Since the relation between electromotive intensity and current is
linear, we are justified in writing the intensities at the adjacent
surfaces of the pair in the form

E.{h') = Z,'I, E.{a") = ZJ'I, (47)

where Zh and Za." depend only upon the material of the conductors
and the geometry of the system. These quantities will be called sur-
face impedances of the inner and outer conductors, respectively.
Inserting (47) in (39) we obtain

A = Z^'I,

^ L-coM - —^1 / log^' + A = - ZJ'I,

(48)

by means of which A and V may be expressed in terms of Z},' and Za".
If we solve the first of these for A and substitute the value thus derived
in the second we get, by virtue of (45),

^' log ^,- = Za" + Z," + io:Le, (49)

27r(g + icoe) ^ h'
or, by (43)

r2 = YZ, (50)

where for brevity we have written

Z = Za" + Z,' + ic^Le. (51)

Direct Conversion of the Circularly Symmetric Field Equa-
tions INTO Transmission Line Equations

As the practical applications of Maxwell's theory become more
numerous, it becomes increasingly important to formulate its exact
connection with transmission line theory. With this purpose in mind,
let us attempt to throw (2) into the form of the transmission line
equations.

The obvious plan of attack is to introduce into (2) the transverse
voltage V and the longitudinal current /, in place of the intensities
E and //. The total current is introduced by substituting (7/2 7rp) for
H^, and the total voltage by integrating the set of equations (2) in
the transverse direction. The first equation gives us nothing of

548 BELL SYSTEM TECHNICAL JOURNAL

importance.'^ The second and third equations, on the other hand,
give

^log^' = E/'(a)-E;(^)-i^, (52)

1 «"

27r(g + icoe) dz

But, upon substituting (45), (47) and (51) in the first of these equations
and (43) in the second, we get

where Z and Y are to be interpreted respectively as the distributed
series impedance and shunt admittance.

Current Distribution in Cylindrical Conductors

So far, we have been dealing with electromagnetic intensities in
dielectrics. We now turn our attention to conductors and determine
their current distributions with the ultimate view of calculating their
surface impedances. One of our sources of information is the familiar
set of equations (12). In these equations, however, we now let e = 0
since the displacement current in conductors is negligibly small by
comparison with the conduction current. From these equations, we
eliminate electromotive intensities and thus obtain a differential
equation for the magnetomotive intensity. The latter is in fact
equation (6) with only one difference: the exponential factor e~^^ has
been explicitly introduced and cancelled so that the equation has
become

or (54)

d
dp

r 1 d{pH^) 1
P dp

= (a^ - Y^)II,,

dm^

I ^^^-P ^V _ ( 1 p2^/T

dp^

P dp p^

0-2 = gwp.

i = 2irgp.fi.

where

This 0- will be called the intrinsic propagation constant of solid metal.

1^ Our standard practice of neglecting the longitudinal displacement currents
has given us the general rule that 2-irpH^ = / is independent of p. Using this relation
in the first of equations (2.2), we get

(g + i(jie)Ez == 0;

but this merely reflects the fact that g + zwe is very small.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 549

The attenuation and the phase constants are each equal to V irg^if.
The intrinsic propagation constants of metals are large quantities
except at low frequencies as the accompanying table indicates.

Propagation Constant of Commercial Copper

g = 5.800 10= mhos/cm.
H = 0.01257 ixh/cm.

= V^rgju/

100

10,000

1,000,000

100,000,000

0.0

0.1513
0.4785
1.513
15.13
151.3
1513.

On the other hand, r is very small; if air is the dielectric between the
conductors, r is of the order of (l/3)ico 10"^'^. Hence, even at high
frequencies T^ is negligibly small by comparison with a^ and we can
rewrite (54) as follows:

d_
dp

ld_
p dp

{pH,)

a-'IK

(55)

This is Bessel's equation and its solution can be written down at

once ^^ as

H^ = Ah{ap) +BK,{<xp),

(56)

where the functions /i(m) and Ki{u) are the modified Bessel functions
of the first order and respectively of the first and second kind. For large
values of the argument we have approximately

if.(«)=^/f,e-"(l+|^

(57)

^^ It is interesting to note that in the case of a fairly thin hollow conductor whose
inner radius is not too small there exist very simple approximate solutions of (55).
Under these circumstances p varies over such a small range that no serious error is
introduced in treating the factors (1/p) and p in (55) as constants, and the equation
becomes

, ., — cr n,p,
dp-

which is satisfied by the exponential functions e"'' and e"''''. The larger the value
of p and the faster the change in H^ with p, the better is the approximation.

550 BELL SYSTEM TECHNICAL JOURNAL

while for small values

1 u u ^^^^

X,(«)=- + ^log|.

The function Ii(u) becomes infinite and Ki(u) vanishes when u is
infinite. ^'^ When u is zero, Ii(u) vanishes and K\{u) becomes infinite.
The longitudinal electromotive intensity is calculated from the
third equation (12) with the aid of the following rules for differentiation
of modified Bessel functions of any order n:

"3 v-"- J-n) ^ -'n— 1)

d ^^^^

Thus,
where

£, = vLAhiap) - BKoiapn (60)

(61)

For reasons which will appear later, this quantity t] will be called the
intrinsic impedance of solid metal.

The current density is merely the product of the intensity E^ and
the conductivity g.

In a general way the behavior of the functions of zero order is
similar to that of the functions whose order is unity. Thus, for large
values of the argument, ,

Uiu) = -^ ( 1 + ^
V27rM \ 8m

(62)

and for small values

h{ii) = 1 - -4 '

(63)
Kq{u) = - log M + 0.116.

" This statement is correct only as long as the real part of u is positive. This is
so in our case because out of two possible values of the square root representing a
we can always choose the one with the positive real part.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 551

Surface Impedance of a Solid Wire

On page 547 we defined the surface impedances of a coaxial pair as
the ratios of the longitudinal electromotive intensities on the adjacent
surfaces of the cylinders to the total currents flowing in the respective
conductors. In that place, however, we were unable to give explicit
formulae for the impedances so defined because we did not yet have a
precise value for E^. Now that this omission has been supplied, we
are prepared to compute Zh and ZJ'.

We consider the case of a solid inner cylinder surrounded by any
coaxial return, and seek to determine the constants A and B in (60).
Since the E.M.I, must be finite along the axis of the wire we must
make B = 0, because the X-function becomes infinite when p = 0.
On the surface of the wire the magnetomotive intensity is Ijl-Kh if
I is the total current in the wire. By equation (56) this intensity
equals AIi{(Th) ; hence,

and the final expression for the electromotive intensity within the
wire is

Thus, we have the following expression for the surface impedance of

the solid wire:

E^ib) r]Io((Tb) , , ,,-v

Zh = — J— = ^ 7 7- / .X , ohms/cm. (65)

1 2ir01i{aO)

As the argument increases, the modified Bessel functions of the
first kind (the /-functions) become more and more nearly proportional
to the exponential functions of the same argument. Thus, if the
absolute value of ab exceeds 50, the Bessel functions in the preceding
equation cancel out and the following simple formula holds within
1 per cent:

^^ = A = ?A \/- (1 + ^)' ohms/cm. (66)

ZTTO Zb \ TTg

This surface impedance consists of a resistance representing the
amount of energy dissipated in heat, and a reactance due to the mag-
netic flux in the wire itself. Separating (66) into these two parts,
we have, approximately,

1 Kf

Rh = o)Lb = :7r\/ — ■

20 \ TTg

552 BELL SYSTEM TECHNICAL JOURNAL

However, most of the error in (66) occurs in the real part. If more
accurate approximations for Bessel functions are used, then

R.=l^f' '

2b\Tg 47rg^>2 '

(67)

these are correct within 1 per cent if | a-& | > 6. The surface inductance
Lb equals {\I^Trh)^ixJTrgf henries/cm.; it decreases as the frequency
increases.

If the wire is so thin or the frequency is so low that \(jb \ < 6,
equation (65) has to be used. Its use in computations is quite simple,
however, because the argument ah is a complex number of the form
mVI; and the necessary functions have been tabulated. Lord Kelvin
introduced the symbols ber u and bei it for the real and the imaginary
parts of loiw^i), so that we now write

Io{u^i) = ber u -\- i bei u. (68)

Differentiating, we have

Vi la'iu^i) — Vi Ii{u^i) = ber' u -\- i bei' u,

and therefore

r-. ber' u -}- i bei' u , .

Ii{u\i) = ^^ (69)

If we insert these values in (65), and recall that the d.-c. resistance
of a solid wire is l/irgb^, and that <r = grj, we obtain at once

Zb ,,,7 u ber u bei' u — bei u ber' m

= TTgb^Zb = -7^

,,^.^0 2 (ber' m)2 + (bei' w)^

, . ti ber u ber' u + bei u bei' u

+ t

(70)

2 (ber' uY + (bei' u)

where u is the absolute value of <xb. The accompanying graph
illustrates the real and imaginary parts of this equation ^^ (Fig. 2).

The Surface Impedances of Hollow Cylindrical Shells ^^
In the case of a hollow conductor whose inner and outer radii are
respectively equal to a and b, the return coaxial path for the current

1^ For equation (70) and various approximations see E. Jahnke and F. Emde.

1* In the case of self-impedances the more general equations of two parallel
cylindrical shells were deduced by Mrs. S. P. Mead. For the special formulae
concerning self-impedances of coaxial pairs see A. Russell.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 553

may be provided either outside the given conductor or inside it or
partly inside and partly outside. We designate by Zaa the surface
impedance with internal return, and by Z^b, that with external return.
These impedances are equal only at zero frequency; but if the con-

d|o

^ y'

y y

^''

< y

/■

3.5 R

::J|c\j

Fig. 2 — The skin effect in solid wires. The upper curve represents the ratio of
the a-c. resistance of the wire to its d-c. resistance and the lower curve the ratio of
the internal reactance to the d-c. resistance.

ductor is thin, they are nearly equal at all frequencies. If the return
path is partly internal and partly external, we have in effect two
transmission lines with a distributed mutual impedance Zah due to
the mingling of the two currents in the hollow conductor common to
both lines. However, since this quantity Zah is not the total mutual
impedance between the two lines unless the hollow conductor is the
only part of the electromagnetic field common to them, it is better to
call Zah the transfer impedance from one surface of the conductor to
the other.

In order to determine these impedances, let us suppose that of the
total current /„ + lb flowing in the hollow conductor, the part /„
returns inside and the rest outside. Since the total current enclosed
by the inner surface of the given conductor is — /„, and that enclosed
by the outer surface is h, the magnetomotive intensity takes the values
— (/a/27ra) and (h/lirb), respectively, at these surfaces. This infor-
mation is sufficient to determine the values of the constants A and B
in the equation (59) governing current distribution. In fact, we

554 BELL SYSTEM TECHNICAL JOURNAL

have

Ah{<xa) + BK.iaa) = - -^ ,

lira

(71)

and therefore

Ah{<jb) +BK,{ab) =^,

ZTTO

. _ Ki(ab) , Ii(aa) j
E = — ^^^^^^ T — ^1^°"^) T

(72)

where

D = h{ab)Ki(aa) - I,(cTa)K,(ab). (73)

Substituting these into the second equation of the set (59), we obtain
the longitudinal electromotive intensity at any point of the conductor.
We are interested, however, in its values at the surfaces since these
values determine the surface impedances. Equating p successively
to a and b, we obtain

(74)

Ezia) — Zaala + Z abib,
Ez{b) = Zbala + Zbhih,

where 2°

Zaa = ^:^\:io{<Ta)K,(ab) + Ko((ra)h(abn

Zbb = 2^ Uo(<rb)K^{aa) + Koi<xb)U(aan (75)

^ab ^^ ^ba

2 TTgabD

The results embodied in equation (74) can be stated in the following
two theorems:

Theorem 1 : If the return path is wholly external {la = 0) or wholly in-
ternal (lb = 0), the longitudinal electromotive intensity on that
surface of a hollow conductor which is nearest to the return path
equals the corresponding surface impedance per unit length multiplied
by the total current flowing in the conductor; and the intensity on the
other surface equals the transfer impedance per unit length multiplied
by the total current.
"" To obtain the last equation, It is necessary to use the identity

Io(.x)Ki(x) + Ko{x)Iiix) = - •

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 555

Theorem 2: If the return path is partly external and partly internal the
separate components of the intensity due to the tivo parts of the total
current are calculated hy the above theorem and then added to obtain
the total intensities.
At high frequencies, or when the conductors are very large, (75) can
be replaced by much simpler approximate expressions.^^ If, however,
we are compelled to use the rigorous equations in numerical computa-
tions, it is convenient to express the Bessel functions in terms of
Thomson functions. Two of these, the ber and bei functions, or
Thomson functions of the first kind, have already been introduced.
The functions of the second kind are defined in an entirely analogous
fashion as

Kq{x4i) = ker x -\- i kei x. (76)

Differentiating, we have

^l^ Ko{x4l) = — 41 Ki{x\[l) = ker' x + i kei' x, (77)

so that

T^ / Px ker' X -\- i kei' x ,^„.

Kiix^lt) = '-^, (78)

All these subsidiary functions have been tabulated ;^^ but the
process of computing the impedances is laborious nevertheless.

The Complex Poynting Vector ^3
In the preceding sections we have been able to determine the surface
impedances of the coaxial conductors by reducing the field equations
to the form of transmission line equations, and interpreting various
terms accordingly. However, if the conductors are eccentric or of
irregular shape, the effective surface impedances are more conveniently
calculated by the use of the modified Poynting theorem.

This theorem states that, if E and H are the complex electromotive
and magnetomotive intensities at any point, and if £* and H* are
the conjugate complex numbers, then ^^

r r lEH*']dS = g f C C {EE*)dv + ico/x f f f {HH*)dv. (79)

^1 See portion of this text under the heading "Approximate Formulae for the
Surface Impedance of Tubular Conductors," page 557.

22 British Association Tables, 1912, pp. 57-68; 1915, pp. 36-38; 1916, pp. 108-122.

23 For an early application of the Complex Poynting vector see Abraham v.
Foppl, Vol. 1 (Ch. 3, Sec. 3).

2^ The brackets signify the vector product and the parentheses the scalar product
of the vectors so enclosed. The inward direction of the normal to the surface is
chosen as the positive direction. The division by 47r does not occur if the consistent
practical system of units is used as it is done in this paper.

556 BELL SYSTEM TECHNICAL JOURNAL

To get an insight into the significance of this equation, let us con-
sider a conductor which is part of a single-mesh circuit, and extend our
integrals over the region occupied by this conductor. Then the first
integral on the right of (79) represents twice the power dissipated in
heat in the conductor, while fxf S S{HH*)dv is four times the average
amount of magnetic energy stored in it.

On the other hand, when we look at the conductor from the stand-
point of circuit theory, these two quantities are respectively RP and
LP; R and L being by definition the "resistance" and "inductance"
of the conductor. Hence we have the equation,

lEH*']ndS = iR + iooL)P = ZP, (80)

from which the impedance Z can be computed when the field intensities
are known at the surface of the conductor.

If, on the other hand, the conductor is part of a two-mesh circuit
and /i and I2 are the amplitudes of the currents in meshes 1 and 2
respectively, the average amount of energy dissipated in heat per
second can be regarded as made up of three parts, two of which are
proportional to the squares of these amplitudes, while the third is
proportional to their product. The first two of these parts being
dependent on the magnitude of the current flowing in one mesh only
are attributed to the self-resistance of the conductor to the corre-
sponding current; the third part is attributed to the mutual resistance
of the conductor. Designating the self-resistances by Rn and i?22 and
the mutual resistance by i?i2, we represent the energy dissipated in
heat in the form l/liRiJi^ + 2R12I1I2 + R^^H)- Similarly, the
average amount of energy stored in the conductor can be represented
in the form l/4(Lii7i2 + 2L 12/1/2 + Li^I^), where Lu and L22 are
called respectively s el j -inductances and L12 mutual inductance. In this
case, equation (79) can be written as follows:

[E//*]n^5 = Zn/x2 + 2Z12/1/2 + Z22/2^ (81)

where the quantities Zw, Z22 and Z12 are respectively the self-im-
pedances and the mutual impedance of the conductor.

In general, if the conductor is part of a ife-mesh circuit, we can
obtain all its self-and mutual impedances by evaluating the integral
X X[EH*\dS over its surface, and picking out the coefficients of
various combinations of /'s.

We shall have an occasion to apply these results in computing the
eff^ect of eccentricity upon the resistance of parallel cylindrical con-
ductors.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 557

Approximate Formula for the Surface Impedance of
Tubular Conductors

The exact formulae (75) for the internal impedances of a tubular
conductor are hard to use for numerical computations, but simple
approximations can be easily obtained if the modified Bessel functions
are replaced by their asymptotic expansions and the necessary division
performed as far as the second term. Thus, we have

Zhh —

2 Tb

lira

coth 0-^ + 7?" ( ~ + T
lex \a b

TT /3 , 1

coth 0-^ — ;i— T + -

2a \ b a

(82)

Zah = -;= csch at^

2Tr^ab

where t is the thickness of the tube,
parts, we have

Separating the real and imaginary

Rbb =

Raa = 7^

Rab = |— r

CjLbb =

^Laa —

CoLab =

2a
1

|Z„J =

'/x/ sinh M + sin M o + 36

TTg cosh u — cos M
ixf sinh u + sin u

leirgab'^ '
b + 3a

TTg cosh w — cos u 16 TTgba^ '

. . u u . , u . u
r-? smh - cos 7; + cosh - sm -^
ixj Z 1 2 2

TTg cosh u — cos u

jjif sinh u — sin u

TTg cosh u — cos u '

fjif sinh u — sin m

Tg cosh u — cos u '

. , M u . u . u

. smn - cos -x — cosh - sm -;r

^ab \ TTg

cosh u — cos zf

V7?

Vrga^ (cosh u — cos i<)

(83)

where u = /V2gco;u.

It is obvious that in the equations for the self-resistances, the second
terms represent the first corrections for curvature and vanish altogether
if the conductors are plane. Although these formulae were derived by
using asymptotic expansions which are valid only when the argument
is large, i.e., at high frequencies, the results are good even at low

558 BELL SYSTEM TECHNICAL JOURNAL

frequencies, provided the tubular conductor is not too thick. Thus,
if the frequency is 0, the first term in the above expression for Rw)
becomes IjlTgbt which is the d.-c. resistance of the tube if its curvature
is neglected. The second term only partially corrects for curvature,
the error being of the order of f/Sb"^. Hence, if the thickness of the
tube is not more than 25 per cent of its high-frequency radius, that is,
the radius of the surface nearest the return path, the error is less than
1 per cent. The formula for the mutual impedance is exceedingly
good down to zero frequency for all ordinary thicknesses.

If the frequency is very high, further approximations can be made
and the formulae simplified as follows:

P 1 [i4 , a-\-3b

Kbb =

Raa
Rab

Ib^Tg ' \6Tvgat)''

_ 1 [i4 b + Za

2a\Trg IGirgba^'

= 4i /^g-(u/2) cos ( 71 -

^ab \ TTg V

~ 4

1 p

2b'\lTg'

2a \irg

yjab \ TTg \ 4

iab NfTTg

If the ratio of the diameters of the tube is not greater than 4/3,
then we have the following formula for the surface transfer impedance:

17.1

(85)

■^d-c- Vcosh U — COS Iv

which is correct to within 1 per cent at any frequency. This ratio is
illustrated in Fig. 3. The ratios of the mutual resistance and the
mutual reactance to the d.-c. resistance are shown in Fig. 4.
In the case of self-resistances, we let

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 559

where r is the high frequency radius of the tube. Thus i?o is the d.-c.
resistance of the tube if the curvature is neglected. Then we have
approximately

u sinh u + sin u t
2 cosh u — cos u Ir '

(87)

if the tube is fairly thin. The curvature correction is positive if the

0.7
^0.6

1
I

0.2

0.1

Fig. 3 — The transfer impedance from one surface of a cylindrical shell to the other.
The curve represents its ratio to the d-c. resistance.

return path is external, and negative if it is internal. The graph of
the first term is shown in Fig. 5.

An interesting observation can be made at once from the formulae
(83) for the self-resistances of a tubular conductor. If the frequency
is kept fixed and the thickness of the conductor is increased from 0,
its resistance (with either return) passes through a sequence of maxima
and minima.^^ The first minimum occurs when w = tf, i.e., when

2* The general fluctuating character of this function was noted by Mrs. S. P.
Mead [12].

560

BELL SYSTEM TECHNICAL JOURNAL

t = \Tr I {2^ gixf) \ the first maximum occurs when u = 2r, etc. This
fluctuation in resistance is due to the phase shift in the current density
as we proceed from the surface of the conductor to deeper layers.
The "optimum" resistance is Roiiir/l) tanh irJ2) = 1.44i?o, plus or

0.8

0.5

3|f\J

0.2

-0.3

-0.8

/
/
/

\
\

/
/

/
/■ —

1
1
1

\
\

/
/

5 6

0.3

:J|fvi

:i|w

=Jh

d|(M

•0.8

Fig. 4 — The ratios of the transfer resistance and transfer reactance of a cylindrical
shell to its d-c. resistance.

minus the curvature correction tllr. If curvature is disregarded,
the ratio of the optimum resistance to the resistance of the infinitely
thick conductor with the same mternal diameter as the hollow con-

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 561

ductor is tanh 7r/2 = 0.92. When u = lir, the ratio reaches its
first maximum coth x = 1.004. At 1 megacycle the optimum thick-
ness of a copper conductor is about 0.1038 mm.

By a method of successive approximations, H. B. Dwight has ob-

3.0

.o2.0

o
u

-( 1.5

X
(0

O 1.0

fVJ

z
+

I
z

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Fig. 5 — The skin effect in cylindrical shells. The curve represents the ratio of the
a-c. resistance of a typical shell to its d-c. resistance.

tained the impedance of a tubular conductor with an external coaxial
return. 2^ His final results appear as the ratio of two infinite power
series, which converge for all values of the variables involved, though
they can be used advantageously in numerical computations only
when the frequencies are fairly low and the convergence is rapid.
We shall merely indicate how Dwight's formula and other similar
formulae can be obtained directly from the exact equations (75).

Let us replace the outer radius h of (75) by o + ^, where / is the
thickness of the wall, and replace the various Bessel functions by their
Taylor series in t\

hiab) = Ioi<Ja + at) = Z

n=o n\

/o^-K^a),

Ko(ab) = Ko(aa + at)

n=o n\

(<rty

(88)

hicb) = lo'(ab) = E^^/o^-^'Ko-a),

n=0 W!

- K,{<rb) = Ko'iab) = t i^Ko^n+i)(^^a)

2" "Skin Effect in Tubular and Flat Conductors," A. I. E. E. Journal, Vol. 37
(1918), p. 1379.

562

BELL SYSTEM TECHNICAL JOURNAL

We thus obtain

Zbh =

It] w=o n\

where ^ „ is defined as

(89)

(90)

In spite of the complicated appearance of (90) the A 's are in reality
ver}' simple functions of aa, as the accompanying list (91) will show.^^

An =

At = h ,

^1 = 0, A2
A,= -

3/7 0

2_ _
2^2

A,= -

r2/,2 '

a'a

a^a"

(91)

. _ 1 , 9 ,60

Aq — - H r— ; + c =

0-a a%^ a^a°

The formula (90) can be made more rapidly convergent by partially
summing the numerator and the denominator by means of hyperbolic
functions. Thus, the numerator becomes

a 1 , , smh at

-r cosh at -\ ;r

0 laa

3_i +
4a ^

_ 3^2

a2 ^

and the denominator

T sinh at +

3(o-/ cosh at — sinh cr^)

The reader will readily see that there would be no difficulty in using
this method to obtain other expansions somewhat similar to (89).
For example, we might write a = b — t in (75) and express our
results in terms of the outer radius. In this respect the method that
we have used has greater flexibility than Dwight's ; but there seems to
be little advantage gained from it, since the simple formulae (82) are
sufficient for most practical purposes.

-^ The values given in (91) are exact, not approximate. One of them, namely,

h'{cra) loicra)

Ko'icra) Koicra)

is one of the fundamental identities found in all books on Bessel functions. The
rest are consequences of analogous, though ess familiar, identities. The general
expressions for the coefficients An were obtained by H. Pleijel [20 J.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 563

Internal Impedances of Laminated Conductors

So far we have supposed that all conductors were homogeneous.
We shall now consider a somewhat more general conductor composed
of n coaxial layers of different substances. As before, we are interested
in finding expressions for the internal impedances; besides, we may
wish to know how the total current is distributed between the different
layers of the conductor.

To begin with, let us suppose that a coaxial return path is provided
outside the given conductor. We number our layers consecutively and
call the inner layer the first. Let Z«a^"'^ and Zbi'^'^'^ be the surface
impedances of the mth layer, the first when the return is internal, the
other when it is external ; and let Zgh^'^^ be the transfer impedance
from one surface to the other. Formulae for these impedances have
already been obtained in the section under "The Surface Impedances
of Hollow Cylindrical Shells," page 552. Also, let z^-^^'^^ be the surface
impedance of the first m layers with external return ; that is, the ratio
of the longitudinal electromotive intensity at the outer surface of the
mth. layer to the total current Im in all m layers. ^^

By hypothesis, there is no return path inside the laminated con-
ductor as a whole. Hence, when we fix our attention on any one
layer alone, say the mth, we may say that the current in this layer
returns partly through the m — \ layers within it, and partly outside.
In the w — 1 inner layers, however, the current is assumed to be
Im-i in the outward direction — or what amounts to the same thing
— Im-i in the return direction. Hence we conclude that, of the
current Im — Im-\ in the layer under discussion, Im returns outside
and — Im-i inside. Substituting these values in Theorem 2 on
page 555, we find that the electromotive intensity along the inner
surface of the layer is Zab^'^^Im — ZaJ-'^^Im~\-

But the inner surface of the wth layer is the outer surface of the
composite conductor comprising the m — \ inner layers, and by
Theorem 1, the electromotive intensity on this outer surface is
Z66^"'~^^/m-i. As the two must be equal, we obtain an equation from
which we can determine the ratio of the current flowing in the first
m — \ layers to that flowing in m layers. This is

h^ ^ ^I^lI . . (92)

In this formula for the effect of an extra layer on the current dis-

28 In this notation, the current flowing in the mth layer is /,„ — /,„_i. It should
also be noted that Zh6<i> = zw'^^

564 BELL SYSTEM TECHNICAL JOURNAL

tribution, it will be noted that the denominator is the impedance
(with internal return) of the added layer plus the original impedance.
We now consider the electromotive intensity on the outer surface
of the mth. layer, which is zih^^'Um on the one hand, and {Zbb^"'^Im
— Zab^"'^Im-i) on the other. Thus, we have the following equation,

expressing the effect of an additional layer upon the impedance of the
conductor.

This equation is a convenient reduction formula. Starting with
the first layer (for which 255 ^^^ = Ztb^^^), we add the remaining layers
one by one and thus obtain the impedance of the complete conductor
in the form of the following continued fraction:

Zaa'^' + Z,5^"-l) + Z„„("-^> + Z,,("-2) + ^^^)

^ aa I ^b

(1)

We can also get a reduction formula for the transfer impedance
between the inner and outer surfaces of the composite conductor
formed by the first m layers. To do so, it is only necessary to note
that, since the inner surface of the first m — 1 layers is also the inner
surface of the first m layers as well, the electromotive intensity on
that surface can be expressed either as Sab^'"~^^/m-i or as Zab'-^'^Im-
Thus, we have

By noting that Zab^^'^ = Zab^^\ we can determine successively the
transfer impedances across the first two layers, the first three, and
so on. This formula is not quite as simple as (94), owing to the
presence of Z66^'"~^> in its denominator, and it is therefore not expedient
to evaluate Zab^""^ explicitly; but it is not prohibitively cumbersome
from the numerical standpoint when the computations are made step
by step.

Although in deducing equations (93) and (95) we supposed that the
added layer was homogeneous, the equations are correct even if this
layer consists of several coaxial layers, provided Zaa^"''^^'' and Zafi^™"^^^
are interpreted as the impedances of the added non-homogeneous
layer in the absence of the original core of m layers. These latter

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 565

impedances themselves have to be computed by means of equations
(94) and (95).

If the return path is inside the laminated conductor, instead of
outside, formulae (92) and (93) still hold, provided we interchange
a and h, and count layers from the outside instead of the inside, so
that w = 1 is the outermost, rather than the innermost, layer.

The basic rule for determining the surface impedances of laminated
conductors can be put into the following verbal form:

Theorem 3: Let two conductors, both of zvhich may be made up of coaxial
layers, fit tightly one inside the other. Any surface self-impedance
of the compound conductor equals the individual impedance of the
conductor nearest to the return path diminished by the fraction
whose numerator is the square of the transfer impedance across this
conductor and ivhose denominator is the sum of the surface impedances
of the two component conductors if each is regarded as the return
path for the other. The transfer impedance of the compound con-
ductor is the fraction whose numerator is the product of the transfer
impedances of the individual conductors and whose denominator is
that of the self -impedance.

If two coaxial conductors are short-circuited at intervals, short
compared to the wave-length, the above theorem holds even if the
conductors do not fit tightly one over the other, provided we add in
the denominators a third term representing the inductive reactance of
the space between the conductors.

Disks as Terminal Impedances for Coaxial Pairs

So far we have been concerned only with infinitely long pairs. We
now take up a problem of a different sort; namely, the design of a
disk which, when clapped on the end of such a pair, will not give rise
to a reflected wave.

The line of argument will be as follows: To begin with, we shall
assume a disk of arbitrary thickness h, compute the field which will
be set up in it, and then adjust the thickness so as to make this field
match that which would exist in the dielectric of an infinite line.

The field in the disk has to satisfy equation (2) where iwe can be
disregarded by comparison with g. Thus, we have

dH, _ 1 {pH,) _

"' ' '" (96)

dE^ _ dEp
dp dz

icofiH^.

d{pH,)

— 0'

H,p ^

566 BELL SYSTEM TECHNICAL JOURNAL

In the dielectric between the coaxial conductors, the longitudinal
displacement current density is very small; in fact, it would be zero
if the conductors were perfect. This current density is continuous
across the surface of the disk and, therefore, gE^ is exceedingly small.
Hence, the second of the above equations becomes approximately

(97)

so that

(98)

where P is independent of p but may be a function of z. Under these
conditions, the remaining two equations are

^ = - i.^H,, ^= - gE,. (99)

From the form of these equations and from (98), we conclude that
the general expressions for the intensities in the disk are

^ Ae- + Be- ^ o{_Be^" - Ae-q

P gP

where a = Vgco/^i.

On the outside flat surface of the disk (given by z = ^ where h is
the thickness of the plate), the magnetomotive intensity is very nearly
zero; ^^ therefore,

Ae''^ + Be-"'' = 0. (101)

From this we obtain

A = - Ce-''\ B = Ce''\ (102)

where C is some constant. Thus equations (100) can be written as
follows :

C sinh a{h — z)

P
P _ oC cosh aiji — z)
gP

(103)

and at the boundary between the disk and the dielectric of the trans-
mission line {z = 0), we have

^=-coth<jh. (104)

J^v g
^" On account of the negligibly small longitudinal current in the disk.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 567

O^n the other hand, if there is to be no reflection this must equal
fj.Je by equation (24). Hence

-coth ah = J-- (105)

If ah is small, coth ah equals approximately I /ah, and •

A=^^icm. (106)

Under these conditions, the generalized flux of energy across the
inner surface of the disk is, in accordance with the text under "The
Complex Poynting Vector," page 555, and equation (14),

0 Jb'

E,H^*p dpd<p =-^ J^ log ^ P. (107)

Thus, the impedance of this disk is a pure resistance equal to the
characteristic impedance of the coaxial pair.

Cylindrical Waves and the Problem of Cylindrical Shields ^^
It is well known that when two transmission lines are side by side,
to a greater or lesser extent they interfere with each other. This
interference is usually analyzed into "electromagnetic crosstalk" and
' ' electrostatic crosstalk. ' '

Thus, electric currents in a pair of parallel wires produce a magnetic
field with lines of force perpendicular to the wires. These lines cut
the other pair of wires and induce in them electromotive forces, thereby
producing what is usually called the "electromagnetic crosstalk";
this crosstalk is seen to be proportional to the current flowing in the
first pair. The "electrostatic crosstalk," on the other hand, is caused
by electric charges induced on the wires of the second system; these
charges are proportional to the potential difference existing between
the wires of the "disturbing" transmission line.

The distinction between two types of crosstalk is valid, although
the terminology is somewhat unfortunate; the word "electromagnetic"
is used in too narrow a sense and the word "electrostatic" is a con-
tradiction in terms since electric currents and charges in a transmission
line are variable. The terms "impedance crosstalk" and "admittance
crosstalk" would be preferable because the former is due to a dis-
tributed mutual series impedance between two lines and the latter is
produced by a distributed mutual shunt admittance.

^^ Since this paper was written, a related paper has been published by Louis V.
King [18]. However, the physical picture here developed appears to be new.
The earliest writer who treated the problem of electromagnetic shielding is H.
Pleijel [21].

568 BELL SYSTEM TECHNICAL JOURNAL

The crosstalk between two parallel pairs (this applies to twisted
pairs as well) can be reduced by enclosing each pair in a cylindrical
metallic shield. It is the object of this and the following two sections
to develop a theory for the design of such shields.

This theory is based upon an assumption that in so far as the radial
movement of energy toward and away from the wires is concerned we
can disregard the non-uniform distribution of currents and charges
along the length of the wires. No serious error is introduced thereby
as long as the radius of the shield is small by comparison with the wave-
length. The field around the wires is considered, therefore, as due to
superposition of two two-dimensional fields of the types given by
equations (4) and (5).

The actual computation of the effectiveness of a given shield will
be reduced to an analogous problem in Transmission Line Theory.

Equations (4) and (5) are too general as they stand. Strictly
speaking the effect of a shield upon an arbitrary two-dimensional
field cannot be expressed by a single number. The field at various
points outside the shield will be reduced by it in different ratios.
However, any such field can be resolved into "cylindrical waves,"
each of which is reduced by the shield everywhere in the same ratio.
Moreover, to all practical purposes the field produced by electric
currents (or electric charges) in a pair of wires is just such a pure
cylindrical wave.

Since both E and H are periodic functions of the coordinate <p,
they can be resolved into Fourier series. The name "cylindrical
waves" will be applied to the fields represented by the separate terms
of the series. As the name indicates the wave fronts of these waves
are cylindrical surfaces, although owing to relatively low frequencies
and long wave-lengths used in practice the progressive motion of
these waves is not clearly manifested except at great distances from
the wires.

Turning our attention specifically to magnetic cylindrical waves
of the nth order, and writing the field components tangential to the wave
fronts in the form E cos tup and H cos n(p, we have from equations (5) :

dE . ^^ d{pH)

dp dp

From these we obtain

{g -f icoe)p +

twjxp

E. (108)

d'^F dE

'j^+ P^- pcoM(g + icoe)p2 + n^-}E. (109)

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 569

This equation, being of the second order, possesses two independent
solutions : one for diverging cylindrical waves and the other for reflected
waves. The ratio oi E to H in the first case and its negative in the
second will be called the radial impedance offered by the medium to
cylindrical waves.

In the next section we shall determine radial impedances in di-
electrics and metals and show that for all practical purposes the
attenuation of cylindrical waves in metals is exponential. The sig-
nificance of the radial impedance is the same as that of the charac-
teristic impedance of a transmission line. When a cylindrical wave
passes from one medium into another, a reflection takes place unless
the radial impedances are the same in the two media. Thus if Eq
and Hq are the impressed intensities (at the boundary between the two
media), Et and Hr the reflected and Et and Ht, the transmitted
intensities, we have

E, + Er = Et and //o + Hr = Ht, (110)

since both intensities must be continuous. On the other hand, if k is
the ratio of the impedance in the first medium to that in the second,
then equations (110) become

kHo - kHr = Ht and Ho + Hr = Ht. (Ill)

Solving we obtain

2k 1

Ht = ^^^^Ho and Et = ^^ ^o- (112)

The reflection loss will be defined as

R = 20 logiof^ decibels. (113)

When a wave passes through a shield, it encounters two boundaries
and if the shield is electrically thick, that is, if the attenuation of the
wave in the shield is so great that secondary reflections can be dis-
regarded without introducing a serious error, the total reflection loss
is the sum of the losses at each boundary. The first loss can be com-
puted directly from (112) and the second from the same equation if
we replace k by its reciprocal. Thus, the total reflection loss for
electrically thick shields is

R = 20 logio-^-^y^' decibels, (114)

570 BELL SYSTEM TECHNICAL JOURNAL

When the ratio of the impedances is very large by comparison with
unity, the formula becomes

i? = 20 1ogio-^, (115)

and when k is very small, then

i? = 20 1ogioj|^. (116)

In the next section we shall see that to all practical purposes, the
wave in the shield is attenuated exponentially. If a. is the attenuation
constant in nepers and if t is the thickness of the shield, then the
attenuation loss is

A = 8.686«^ decibels (117)

and the total reduction in the magnetomotive intensity due to the
presence of the shield is

S = R-\-A. (118)

The electromotive intensity is reduced in the same ratio.

But if the shield is not electrically thick, a correction term has to
be added to the reflection loss. This correction term can be shown
to be 31

{k - 1)2 ^_

C = 20 logi

decibels, (119)

{k + 1)2^

and if k is very large or very small by comparison with unity then

C = 3 - 8.686a/ + 10 logio (cosh 2at - cos 2/3/). (120)

Equation (120) does not hold down to / = 0; when Tt is nearly zero,
then

{k - 1)2

. C = 20 logi

{k + 1)^

(121)

So far we Supposed that the shields were coaxial with the source.
If this is not so, it is always possible to replace any given line source
within the shield by an equivalent system of line sources coaxial with
the shield and emitting cylindrical waves of proper orders. Mathe-
matically this amounts to a change of the origin of the coordinate
system. In the next section we shall see that the shielding effective-
ness is not the same for all cylindrical waves. This means, of course,
that if the shield is not coaxial with the source, the total reduction in

^1 Here, r = a -\- ip \s the propagation constant in the shield.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 571

the field depends upon the position of the measuring apparatus. The
variation is very small, however, unless the source is almost touching
the shield and it can be stated that approximately the shielding
effectiveness is independent of the position of the source.

It is interesting to observe from the accompanying tables that
while the attenuation loss is greater in iron than in copper, the reflection
loss is greater at a copper surface. In fact, at some frequencies the
impedances of iron and air nearly match and practically no reflection
takes place. Hence, a thin copper shield may be more effective than
an equally thin iron shield. And if a composite shield is made of
copper and iron, the shield will be more effective if copper layers are
placed on the outside to take advantage of the added reflection.

TABLE I

The Absolute Value of the Radial Impedance Offered by Air to Cylindrical
Magnetic Waves of the First Order (in Microhms)

1 cycle

10 cycles. . . .

100 cycles. . . .

1 kilocycle . .

10 kilocycles.

100 kilocycles. ,

1 megacycle .

10 megacycles

100 megacycles

Radius = 0.5 cm.

1 cm.

0.0395
0.395
3.95
39.5
395.
3,950.
39,500.
395,500.

3.95 ohms

0.07896
0.790
7.90
79.0
790.
7,900.
79,000.
790,000.

7.9 ohms

0.1579
1.58
15 8
158.
1,580.
15,800.
158,000.

1.58 ohms
15.8 ohms

TABLE II
The Intrinsic Impedance of Certain Metals (r;)/(V«) in Microhms

Copper

Lead

Aluminum

Iron

g = 5.8005 X 105

g = 4.8077 X 10*

g = yw "

g = 10= mhos/cm.

mhos/cm.

ralios/cm.

M = 1.257 filijcm.

M = 0.01257 M^i/cm.

M = 0.01257 ju/z/cm.

M = 0.01257 ,i/z/cm.

= ( 100 relative
to copper)

1 cycle

0.369

1.28

0.487

8.88

10 cycles

1.17

4.05

1.54

28.1

100 cycles

3.69

12.8

4.87

88.8

1 kilocycle. . . .

11.7

40.5

15.4

281.

10 kilocycles . . .

36.9

128.

48.7

888.

100 kilocycles . . .

117.

405.

154.

2,810.

1 megacycle . .

369.

1,280.

487.

8,880.

10 megacycles .

1,170.

4,050.

1,540.

28,100.

100 megacycles .

3,690.

12,800.

4,870.

88,800.

572

BELL SYSTEM TECHNICAL JOURNAL

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ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 573

Cylindrical Waves in Dielectrics and Metals

In good dielectrics g is small by comparison with we and the first
term on the right in (109) very nearly equals (lirp/xy where X is the
wave-length. But we are interested in wave-lengths measured in
miles and shields with diameters measured in inches;, thus we shall
write (109) in the following approximate form:

When n ^ 0 there are two independent solutions

El = p-« and E^ = p""; (123)

and when w = 0,

El = log p and £2 = 1. (124)

The corresponding expressions for H are, by (108),

Hi = "^^ and H2= - ^ . (125)

in the first case, and

Hi = J— and Hi = 0, (126)

Iwfxp

in the second.

The second case in which Ei and Hi are the electromotive and mag-
netomotive intensities in the neighborhood of an isolated wire carrying •
electric current is of interest to us only in so far as it helps to interpret
(123) and (125). If we were to consider 2n infinitesimally thin wires
equidistributed upon the surface of an infinitely narrow cylinder, the
adjacent wires carrying equal but oppositely directed currents of
strength sufficient to make the field different from zero, and calculate
the field, we should obtain expressions proportional to Ei and Hi.
An actual cluster of 2n wires close together would generate principally
a cylindrical wave of order n; the strengths of other component waves
of order 3w, 5w, etc. rapidly diminish as the distance from the cluster
becomes large by comparison with the distance between the adjacent
wires of the cluster. For the purposes of shielding design we can
regard a pair of wires as generating a cylindrical wave of the first
order (w = 1). The radial impedance of an wth order wave is

^^ Hi~ n ' ^^^^^

574 BELL SYSTEM TECHNICAL JOURNAL

and that of the corresponding reflected wave has the same value.
It should be noted that by the "reflected" cylindrical wave in the
space enclosed by a shield, we mean the sum total of an infinite number
of successive reflections. Each of the latter waves condenses on the
axis and diverges again only to be re-reflected back ; in a steady state
all these reflected waves interfere with each other and form what
might be called a "stationary reflected wave." Not being interested
in any other kind of reflected waves we took the liberty of omitting
the qualification.

In conductors the attenuation of a wave due to energy dissipation
is much greater (except at extremely low frequencies) than that due
to the cylindrical divergence of the wave. Hence, in the shield we
can regard the wave as plane and write (108) in the following approxi-
mate form:

f = - -''^- "f - - «^- (128)

In form, these are exactly like ordinary transmission line equations.
Hence, in a shield the radial impedance is simply the intrinsic im-
pedance of the metal,

Zp = r? = J^ohms, (129)

and the propagation constant,

0- = \'zco/ig = ^irfiigi]. + i) nepers /cm. (130)

The exact value of the radial impedance in metals can be found by
solving (108). Thus, we can obtain

for diverging waves, and

for the reflected waves.

Cylindrical waves of the electric type can be treated in the same
manner. It turns out that the transmission laws in metals are identical
with those for magnetic waves. The radial impedance in perfect
dielectrics, on the other hand is given by

Zp = ^ . (133)

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 575

This is enormous by comparison with the impedance in metals,
thereby explaining an almost perfect "electrostatic" shielding offered
by metallic substances. Even when the frequency is as high as 100 kc.
the radial impedance of air 1 cm. from the source is about 36 X 10^
ohms while the impedance of a copper shield is only 117 X 10~® ohms.
The reflection loss is approximately 220 db.

Power Losses in Shields

As we have shown in the text under "The Complex Poynting
Vector," page 555, the average power dissipated in a conductor is the
real part of the integral <J> = 1/2 J^ J' \^EH*2ndS taken over the surface
of the conductor. If the source of energy is inside a shield, the
integration need be extended only over its inner surface, because
the average energy flowing outward through this surface is almost
entirely dissipated in the shield, the radiation loss being altogether
negligible. If a cylindrical wave whose intensities at the inner
surface of the shield of radius "a" are

H^ = Hq cos n(p, Hp = Ha sin n(p, E, = r]H^, (134)

7] = iwixajn being the radial impedance in the dielectric, is impressed
upon the inner surface of the shield, a reflected wave is set up. The
resultant of the magnetomotive intensities in the two is readily found
to be {Ikjk + \.)Ho, where k is the ratio of the radial impedance of
the dielectric column inside the shield to the impedance Z looking
into the shield. If the shield is electrically thick, the impedance Z is
obviously the radial impedance of the shield; otherwise it is modified
somewhat by reflection from the outside of the shield. The average
power loss in the shield per centimeter of length is, then, the real
part of

2irakk*Z TT TT * /1->r\

* = (k + m' + 1)"'"'- (1^5)

This becomes simply

•I* = iTraZH^H,,*, (136)

if the frequency is so high that k is large as compared with unity.

If the source of the impressed field is a pair of wires along the axis
of the shield, the magnetomotive intensity on the surface of the shield
can be shown to be

H^ = - — •„/ cos (p, (137)

576 BELL SYSTEM TECHNICAL JOURNAL

where I is the separation between the axes of the wires. Therefore,

$= k-^^ p (138)

Resistance of Nearly Coaxial Tubular Conductors

When two tubular conductors are not quite coaxial, a proximity
effect ^^ appears which disturbs the symmetry of current distribution
and therefore somewhat increases their resistance. This effect can
be estimated by the following method of successive approximations.
To begin with, we assume a symmetrical current distribution in the
inner conductor. The magnetic field outside this conductor is then
the same as that of a simple source along its axis and can be replaced
by an equivalent distribution of sources situated along the axis of the
outer conductor. The principal component of this distribution is a
simple source of the same strength as the actual source and does not
enter into the proximity effect. The next largest component is a
double source given by

„ ioiixll .

hz = -r COS d,

// (139)

He = -?i — 5 COS 9,

where / is the interaxial separation, r is the distance of a typical point
of the field from the axis of the outer conductor, and d is the remaining
polar coordinate.

This field is impressed upon the inner surface of the outer conductor ^^
and the resulting power loss equals, by equation (136), the real part of

$ = 2^av (^yP^^, rjP, (140)

where at high frequencies -q = ^iwiijg is simply the intrinsic impedance
of the outer conductor.^^ This loss increases the resistance of the
outer tube by the amount.

^Ra

^#. " (141)

32 For promixity effect in parallel wires external to each other, the reader is
referred to the following papers: John R. Carson [1], C. Manneback [9], S. P.
Mead [12].

33 The radius of this surface is designated by a.

34 At low frequencies j? has to be replaced by the radial impedance looking into
the shield.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 577

He = ;;j-^— ^cos 9,

in excess of the concentric resistance Ra — {lllaj^nflirg given by
(84). The relative increase is, therefore,

The magnetic field (139) is partially reflected from the outer tube,
impressed upon the inner conductor, partially refracted into it and
dissipated there. Using (110) and (1 1 1) we can show that the reflected
field is

ll_
27ra2

^^ = '^ — 2^ P COS 6.

This field converges to the axis of the outer conductor. In order to
estimate its effect upon the inner conductor, it is convenient to replace
it by an equivalent field converging toward the axis of the inner
conductor. By properly changing the origin of the coordinate system
this equivalent field can be shown to be

E, = - y— 2 (/ + P cos if),

■ti^ = — ;s 5 cos (f.

Apptying once more (138) (replacing there a by the radius b of the
inner conductor), we find that the power loss due to this field is given
by the real part of

/)/2

so that the absolute increase in resistance of the inner conductor is

^R.='^J^ (146)

which must be added to the concentric resistance of the inner con-
ductor Rb = {ll2b)yl/xf/Tg. The relative increase is therefore

It is unnecessary to carry the process further.

578 BELL SYSTEM TECHNICAL JOURNAL

Considering the pair as a whole, the resistance when concentric is
R = R„ -\- Rb, and the increase due to eccentricity is AT? = ARa + ARb
thus giving a percentage increase,

It is obvious that, so long as h and / are small compared with a,
this percentage increase is very small.

From the well-known formuhe for the inductance and the capacity
between parallel cylindrical conductors, we find that the characteristic
impedance of a nearly coaxial pair is given in terms of the characteristic
impedance of the coaxial pair by

e2^2

Z = Zo

{k'' - 1) log ^ J'

(149)

where the "eccentricity" e is defined as the ratio of the interaxial
separation to the inner radius of the outer conductor and k as the ratio
of the inner radius of the outer conductor to the outer radius of the
inner conductor. Combining (149) and (148) we have for the attenua-
tion of the nearly coaxial pair:

= ao

1 + 2l> ^'''

k ' (^2 _ 1) log k _

(150)

References
■ Papers

1. John R. Carson, "Wave Propagation Over Parallel Wires: The Proximity

Effect," Phil. Mag., Vol. 41, Series 6, pp. 607-633, April, 1921.

2. John R. Carson and J. J. Gilbert, "Transmission Characteristics of the Sub-

marine Cable," Jour. Franklin Institute, p. 705, December, 1921.

3. John R. Carson and J. J. Gilbert, "Transmission Characteristics of the Sub-

marine Cable," Bell Sys. Tech. Jour., pp. 88-115, July, 1922.

4. John R. Carson, "The Guided and Radiated Energy in Wire Transmission,"

A.I.E.E. Jour., pp. 908-913, October, 1924.

5. John R. Carson, "Electromagnetic Theory and the Foundations of the Electric

Circuit Theory," Bell Sys. Tech. Jour., January, 1927.

6. S. Butterworth, "Eddy Current Losses in Cylindrical Conductors, with Special

Applications to the Alternating Current Resistances of Short Coils," Phil.
Trans., Royal Soc. of London, pp. 57-100, September, 1921.

7. H. B. Dwight, "Skin Effect and Proximity Effect in Tubular Conductors,"

A.I.E.E. Jour., Vol. 41, pp. 203-209, March, 1922.

8. H. B. Dwight, "Skin Effect and Proximity Effect in Tubular Conductors,"

A.I.E.E. Trans., Vol. 41, pp. 189-195, 1922.

9. C. Manncback, "An Integral Equation for Skin Effect in Parallel Conductors,"

Jour, of Math, and Physics, April, 1922.

10. H. B. Dwight, "A Precise Method of Calculation of Skin Effect in Isolated

Tubes," A.I.E.E. Jour., Vol. 42, pp. 827-831, August, 1923.

11. H. B. Dwight, "Proximity Effect in Wires and Thin Tubes," A.I.E.E. Jour.,

Vol. 42, pp. 961-970, September, 1923; Trans., Vol. 42, pp. 850-859, 1923.

ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 579

12. Mrs. S. P. Mead, "Wave Propagation Over Parallel Tubular Conductors: The

Alternating Current Resistance," Bell Sys. Tech. Jour., pp. 327-3vS8, April,
1925.

13. Chester Snow, "Alternating Current Distribution in Cylindrical Conductors,"

Scientific Papers of the Bureau of Standards, No. 509, 1925.

14. John R. Carson and Ray S. Hoyt, "Propagation of Periodic Currents over a

System of Parallel Wires," Bell Sys. Tech. Jour., pp. 495-545, July, 1927.

15. John R. Carson, " Rigorous and Approximate Theories of Electrical Transmission

Along Wires," Bell Sys. Tech. Jour., January, 1928.

16. John R. Carson, "Wire Transmission Theory," Bell Sys. Tech. Jour., April, 1928.

17. A. Ermolaev, "Die Untersuchung des Skineffektes — Drahten mit Complexer

Magnetischer Permeabilitate," Archiv.f. Elektrotechnik, Vol. 23, pp. 101-108,
1929.

18. Louis V. King, "Electromagnetic Shielding at Radio Frequencies," Phil. Mag.,

Vol. 15, Series 7, pp. 201-223, February, 1933.

19. E. J. Sterba and C. B. Feldman, "Transmission Lines for Short-W^ave Radio

Systems," Proc. I. R. E., July, 1932, and Bell Sys. Tech. Joiir., July, 1932.

20. H. Pleijel, "Berakning af Motstand och Sjalfinduktion," Stockholm, K. L.

Beckmans Boktryckeri, 1906.

21. H. Pleijel, "Electric and Magnetic Induction Disturbances in Parallel Conducting

Systems," 1926, Ingeniorsvetenskapsakademiens Handlingar NR 49.

22. J. Fisher, " Die allseitige in zwei Kreiszylindrishen, konaxial geschichteten Stoffen

bei axialer Richtung des Wechselstromes," Jahrbuch der drahtlosen Tele-
graphic und Telephonie, Band 40, 1932, pp. 207-214.

Books

J. Clerk Maxwell, "Electricity and Magnetism," Vols. 1 and 2.

O. Heaviside, "Electrical Papers."

Sir William Thomson, "Mathematical and Physical Papers."

Lord Rayleigh, "Scientific Papers."

Sir J. J. Thomson, "Recent Researches m Electricity and Magnetism."

A. Russell, "A Treatise on the Theory of Alternating Currents."

John R. Carson, "Electric Circuit Theory and the Operational Calculus."

R. W. Pohl, "Physical Principles of Electricity and Magnetism."

Max Abraham and R. Becker, "The Classical Theory of Electricity and Magnetism."

E. Jahnke and F. Emde, "Tables of Functions," B. G. Teubner, 1933.

Note: This list of references is by no means complete. Only the more recent
papers dealing with some phase of the subject treated here are included.

Contemporary Advances in Physics, XXVIII
The Nucleus, Third Part *

By KARL K. DARROW

This article deals first with the newer knowledge of alpha-particle emis-
sion: that common and striking form of radioactivity, in which massive
atom-nuclei disintegrate of themselves, emitting helium nuclei (alpha-
particles) and also corpuscles of energy in the form of gamma-rays or high-
frequency light. There follows a description of the contemporary picture
of the atom-nucleus, in which this appears as a very small region of space
containing various charged particles, surrounded by a potential-barrier; and
the charged particles within, or those approaching from without, are by the
doctrine of quantum mechanics sometimes capable of traversing the barrier
even when they do not have sufficient energy to surmount it. The expo-
nential law of radioactivity — to wit, the fact that the choice between dis-
integration and survival, for any nucleus at any moment, seems to be alto-
gether a matter of pure chance — then appears not as a singularity of nuclei,
but as a manifestation of the general principle of quantum mechanics: the
principle that the underlying laws of nature are laws of probability. More-
over it is evident that transmutation of nuclei by impinging charged par-
ticles, instead of beginning suddenly at a high critical value of the energy of
these particles, should increase very gradually and smoothly with increasing
energy, and might be observed with energy- values so low as to be incompre-
hensible otherwise; and this agrees with experience.

Diversity of Energies in Alpha- Particle Emission

ON EVERYONE who studied radioactivity some twenty years
ago, there was impressed a certain theorem, an attractively
simple statement about the energy of alpha-particles: it was asserted
that all of these which are emitted by a single radioactive substance
come forth from their parent atoms with a single kinetic energy and a
single speed. When beams of these corpuscles were defined by slits
and deflected by fields for the purpose of measuring charge-to-mass
ratio, nothing clearly contradicting this assertion was observed: the
velocity-spectrum of the deflected corpuscles appeared to consist of a
single line. In studying the progression of alpha-particles across
dense matter, it was indeed observed that not all of those proceeding
from a single substance had sensibly the same range. It is, however,
to be expected that if two particles should start with equal energy
into a sheet of (let us say) air or mica, they would usually traverse
unequal distances before being stopped ; for the stopping of either would

* This is the second and concluding section of "The Nucleus, Third Part," begun
in the July, 1934 Technical Journal.

"The Nucleus, First Part" was published in the July, 1933 issue of the Bell Sys.
Tech. Jour. (12, pp. 288-330), and "The Nucleus, Second Part" in the January, 1934
issue (13, pp. 102-158).

580

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 581

be brought about by its encounters with atoms and the electrons which
atoms contain; and there would be statistical variations between the
numbers of atoms and electrons which different particles would
encounter after plunging into such a sheet. The probable effect of
these variations can be computed; and it was shown at an early date
that for at least some of the substances emitting alpha-rays — a~
emitters — the diversity in ranges of the particles is no broader than
should be expected. The curve of distribution-in-range of an a-ray
beam often consists of a single peak or hump, and the shape and
breadth of the hump are consistent with the assumption that it is
due entirely to the "straggling" (the name applied to the statistical
variations aforesaid) of particles all possessed initially of a single
speed.

The vanishing of this beautiful but too-simple theorem from physics
is due to experiments of three types. First, it was found that when
all of the well-known a-particles of about 8.6 cm. range from ThC
were completely intercepted by a stratum of matter of rather more
than 8.6-cm. A.E. (air-equivalent ^2), and the detecting apparatus
was adjusted to a sensitiveness much greater than would have been
tolerable for the main beam, a very few particles — a few millionths
of the number in the 8.6-cm. flock — were still coming through. Some
of these had ranges as great as 11.5 cm., immensely greater than
could be ascribed to straggling. These are the "long-range" alpha-
particles, other examples of which have been discovered with RaC
and (very lately) with AcC. Next the colossal new magnet at
Bellevue near Paris was employed by Rosenblum for deflecting a-
particles and observing their velocity-spectrum, and the unprecedented
dispersion and resolving-power (to employ optical terms) of this
superb apparatus disclosed that for several a-emitters (the list now
comprises eight) the spectrum consists of two or several lines instead
of only one. The "groups" of alpha-particles to which these lines
bear witness lie closer to one another in energy than the aforesaid
long-range particles lie to the medium-range ones, wherefore they are
often said to constitute the "fine-structure" of the alpha-rays; but it
is probable that a more significant basis for distinction lies in the fact
that the long-range corpuscles are relatively scanty, while the various
lines of a fine-structure system are not so greatly unequal in intensity.

'2 1 recall that while the range of an alpha-particle of given speed depends on the
density and nature of the substance which it is traversing, the student is usually
dispensed from taking account of this by the fact that the investigators nearly always
state, not the actual thickness of the actual matter which they used, but the thickness
of air at a standard pressure and temperature (usually 760 mm. Hg and 15° C.)
which would have the same effect.

582 BELL SYSTEM TECHNICAL JOURNAL

Another great magnet of a peculiar and original construction, de-
veloped at the Cavendish, was then applied both to spectra displaying
line structure and to the spectrum of RaC, with notable success;
while the technique of determining distribution-in-range curves has
been improved to such an extent that it now almost rivals the magnets
in its capacity of distinguishing separate groups in an alpha-ray beam.
The theorem of the unique speed is therefore like so many another
theorem of physics; it was valid so long as the delicacy of the experi-
mental methods was not refined beyond a certain point, its validity
ceased as soon as that point was passed.

To enter now into detail :

The long-range particles were discovered by observing scintillations,
a method of singular delicacy and value, but having great dis-
advantages: all the observations being ocular, it is wearisome and
taxing, not every eye is capable of it, and there is no record left behind
except in the observer's memory or notes. Tracks of some of these
particles were later photographed in the Wilson chamber, but it
is a long research to procure even a few hundred of such photographs,
and yet even a few hundred are not sufficiently many for plotting a
really good distribution-in-range curve (the disagreements between
the early work with scintillations and the subsequent work with
Wilson chambers are rather serious). The best available curve is that
which Rutherford, Ward and Lewis obtained with the method of the
"differential ion-chamber," of which the principle is as follows:

When an alpha-particle (or, for that matter, a proton) traverses a
sheet of matter, its ionizing power or ionization per-unit-length-of-
path — we may take one mm. as a convenient unit of air-equivalent —
varies in a characteristic way with the length of path which the particle
has yet to traverse before being stopped completely. The ionization-
curve at first is nearly horizontal, then rises to a pretty sharp maximum,
then falls rapidly to zero.^^ Suppose now that the particle traverses a
pair of shallow ionization-chambers, each containing a gas of which
the thickness amounts to not more than a few mm. of air-equivalent
(the same for both) and the two separated by a metal wall of negligible
air-equivalent. Suppose further that the metal wall is both the
negative electrode of the one chamber and the positive electrode of
the other, and that it is connected to the electrometer or other de-
tecting device. The charge which is perceived is then the difference
between the ionizations in the two chambers. If these are traversed
by a particle which is yet far from the end of its range, the difference

"The curve for protons is exhibited in Fig. 7 of "The Nucleus, Second Part,"
p. 124.

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 583

will be small and perhaps Imperceptible; if by a particle which is
approaching its maximum ionizing-power, the difference will be
appreciable and of one sign ; if by a particle which is coming to the end
of its range, the difference will be considerable and of the opposite
sign. So the differential chamber and its detecting device (in these
experiments, an oscillograph connected through an amplifier, reacting
appreciably to the passage of a single particle) are sensitive above all
to particles which are nearing the ends of their ranges; and if a small
number of such corpuscles be mingled with even a much greater
number of faster charged particles — be they alpha-particles, be they
protons, be they the fast electrons produced by gamma-rays— this
circumstance, which would cripple any other method, will be almost
without effect on it.^^ If the readings of the electrometer are plotted
against the air-equivalent of the thickness of matter between the
source (of alpha-particles) and the chamber, the resulting curve should
not be much distorted from the ideal distribution-in-range curve.

The curve obtained in this way for the long-range particles of RaC
exhibits a notable peak at range 9.0 cm. ; to one side thereof a very
much lower hump at smaller range (7.8) ; to the other side a wavy
curve with four distinct maxima, which Rutherford and his colleagues
deem to be the superposition not of four peaks only, but of seven.
I show this portion (Fig. 5) to illustrate the analysis of such a curve
for groups. Even the tallest of the peaks just mentioned is a mere
molehill compared to the mountain which the principal group of RaC
— the 6.9-cm. a-particles which were formerly the only ones known —
would form if it could be plotted on the same sheet of paper; for the
abundances of the 7.8-cm., 9.0-cm. and 6.9-cm. groups stand to one
another as 1 : 44 : 2,000,000.

These, however, are not the latest words concerning the a-spectrum
of RaC. The energies of these groups might be deduced from their
ranges, but for this purpose it is necessary to use an empirical curve
of energy vs. range which at the time of the foregoing spectrum-
analysis had been extended only up to range 8.6 cm. It was desirable
to measure the energies of some of these groups directly, not only for
their intrinsic interest but in order to carry onward that empirical
energy-ZJ.?. -range relation. Recourse must therefore be had to a de-
flection-method.

Now in the usual form of magnet employed in deflection-experiments

the field pervades the whole of the space between the solid disc-shaped

faces of two pole-pieces. Were the pole-pieces to be so hollowed out

" Also a proton near the end of its range can be distinguished from an alpha-
particle near the end of its range, on account of the difference in maximum ionization-
per-unit-length ("The Nucleus, Second Part," pp. 124-125).

584

BELL SYSTEM TECHNICAL JOURNAL

2 l-l

;0.8

'0.7
0.6
0.5
0.4
0.3
0.2
0.1

•

{

I'l
ll

•\

,^^

•1

^ ,

1
1

\\
\v

'■~\

;
/

f '-

1
1

\
I

^1
1

/
/
/

\
\

;

\
\

1
1
1

\ A

1
1

\
\

1
\

\
\

1 \

1
1

1
/

\ 1

V
/\

\
\

i>*-ii —

0 9.2 9.4 9.6

10.0 10.2 10.4 10.6 10.8 1 1.0 11.2
RANGE IN AIR IN CENTIMETERS

11.4 11.6 11.8 12.0 12.2

Fig. 5 — Distribution-in-range of the long-range alpha-particles proceeding from
RaC', determined with the differential ionization-chamber (Rutherford Ward &
Lewis; Froc. Roy. Soc).

that their disc-shaped faces were reduced to narrow circular rings,
there would be a great economy in magnetizable metal and a great
reduction of weight and volume of the apparatus, as well as other
advantages. This need not impair the availability of the magnet for
analyzing a beam of a-particles, provided that the a-emitter can be
located in the narrow annular space between the faces of the rings, and
provided that the magnetization of the metal can be varied sufficiently
widely. For then, for each group of a-particles there will be a certain
value of the field-strength, whereby those particles which start out in
directions nearly perpendicular to the field and tangent to the rings
will be swept around in circular paths which are confined within that
narrow an,nular space where alone the field exists. Somewhere in that
space the detector should be placed; and the curve of its reading vs.
field-strength H should show a peak for every group, and from the
abscissa of the peak and the radius of the rings the energy of the group
may readily be computed.

Such a magnet was built after Cockcrof t's design at the Cavendish ;
the radius of its rings is 40 cm., they are 5 cm. broad and 1 cm, apart
(these figures are the dimensions of the annular space), and the field-
strength was adjustable up to 10,000 (later 12,000) gauss which was

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 585

sufficient for a-particles of energy up to and even beyond 10.6 MEV
(millions of electron-volts) and range up to and even beyond 11.5 cm.
Figure 6 exhibits the outward aspect, Fig. 7 the cross-section of this
device (one sees how the armature is fully contained within the rings).
The annular space and everything within it was evacuated (being
walled in by the ring B seen in the figures); the detector — a simple
ionization-chamber connected through a linear amplifier to an oscillo-
graph— ^was set 180° around the annulus from the source. This
device in the hands of Rutherford, Lewis and Bowden proved itself
able to furnish even a better spectrum than the scheme of the differ-
ential ionization-chamber; all of the peaks indicated by the former
curve were clearly separated, a hump which had suggested two groups
was resolved into three maxima, and an extra group was discovered —
twelve altogether! (Incidentally, the empirical energy-w.-range curve
of a-particles had previously been extended with the same device by
Wynn-Williams and the rest, to energy = 10.6 MEV and range
= 11.6 cm.)

The long-range spectrum of RaC is thus of no mean complexity.
There will be occasion later for quoting its actual energy-values.
Of the long-range spectrum of ThC there is relatively little to be said;
evidently it has not been studied so intensively as the other, but it
seems to be comparatively simple, for only two groups have been
recognized. One of the groups of ThC has about the same range as
the highest group of RaC, so that between them they comprise the
most energetic subatomic particles ever yet discovered (about 10.6
MEV) apart from those of the cosmic rays and those resulting from
certain artificial transmutations. As for AcC, it is one of the con-
stituents of the mixture of radioactive bodies known as actinium
active deposit, from which a-particles of a range of about 10 cm.
have lately been observed in the Institut du Radium.

Thus far I have written as though RaC and ThC were isolable
substances, of which one may obtain pure samples and analyze at
leisure the a-rays thereof. The truth, however, is far otherwise; for
the difficulties of making one radioactive substance practically free
from others, serious in most cases, are utterly insuperable in these.
Both RaC and ThC are so very ephemeral (their half-lives are too
small to measure, and are guessed from the Geiger-Nuttall relation as
10~^ and 10~^^ second respectively) that they can never be dissevered
from their mother-elements RaC and ThC which are also a-emitters.
Sometimes one finds the long-range particles designated as belonging to
RaC or ThC, and indeed I have nowhere found stated any compelling
reason for attributing them to the C-elements rather than the C-

586

BELL SYSTEM TECHNICAL JOURNAL

Fig. 6— Annular magnet employed for analysis of alpha-ray spectra. (After Ruther-
ford, Wynn- Williams, Lewis & Bowden; Proc. Roy. Soc).

Fig. 7 — Cross-sect ion'of the annular magnet used for analysis of alpha-ray spectra.

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII

587

elements, apart from what is known about the correlated gamma-rays
(see footnote 21).

Fine-structure was discovered, as I said before, with the great
magnet of Bellevue. This has soHd pole-pieces (75 cm. in diameter!)
instead of rings; it is not necessary to adjust the field-strength step by
step so as to bring group after group to a narrow detector,; all the groups

o
a 60

°50

140

> ^0

._--

-''

--"'

-J

tE 20

3.3 3.4

3.7 3.8
RANGE IN

3.9 4.0 4.1 4.2

AIR IN CENTIMETERS

Fig. 8 — Alpha-ray spectrum of RaC; peak observed with differential ionization-
chamber, never before detected because of immensely greater number of particles in
RaC peak just off the diagram to the right; asymmetry indicating fine-structure.
(Rutherford, Ward & Wynn- Williams; Proc. Roy. Soc).

-1

UJ o

I- 3

tr O

< a

Q. o

ct z

UJ <

18
16
14
12

8490 8470 8450 8430 8410 8390 8370 8350 8330 8310 8290

MAGNETIC FIELD IN GAUSS

Fig. 9 — Fine-structure of alpha-ray spectrum of RaC; as asymmetric peak of Fig. 8
resolved into two nearly equal peaks by annular magnet. (Rutherford, Wynn-
Williams, Lewis & Bowden; Proc. Roy. Soc).

of various speeds are deviated simultaneously in circular arcs each of
its own particular radius, and simultaneously fall upon a photographic
plate, producing what looks precisely like a line-spectrum in optics
(Figs. 10, 11). The example in Fig. 10 relates to ThC, the earliest to

588

BELL SYSTEM TECHNICAL JOURNAL

be analyzed; four lines only are visible upon the reproduction, but
some plates after long exposure have shown as many as six.* The
fine-structure of AcC consists of a pair of lines, which were detected
as peaks in the distribution-in-range curve obtained at the Cavendish
with a differential ionization-chamber. Instead of showing this curve
I have chosen the corresponding curve for RaC, albeit it shows only a
single hump (Fig. 8).^^ The unsymmetrical shape of this hump, how-

Fig. 10 — Fine-structure of alpha-ray spectrum of ThC (not completely brought out
in picture) obtained with Bellevue magnet. (S. Rosenblum.)

ThC

I I
ThC RaA

Fig. 11 — Alpha-ray spectra of several elements (those of Po and AcC shifted
slightly to the right with respect to the rest). (S. Rosenblum, Origine des rayons
gamma, Hermann & Cie.).

ever, implies that it really consists of a pair of overlapping peaks;
and so it does; for when the Cavendish magnet was applied to an
a-ray beam from this element, the curve of detector-reading vs. field-
strength displayed two equal peaks quite sharply separate (Fig. 9).

According to Rosenblum's latest census (February 2, 1934) there
are now eight known examples of fine-structure: from the radium
series, Ra (two groups), RaC (2); from the thorium series, RdTh (2),
ThC (6) ; from the actinium series, RdAc (no fewer than eleven groups,
the richest case of all!), AcX (3), An (3), AcC (2). According to
Lewis and Wynn-Williams, there are (or were, in the spring of 1932)

* I am indebted to Dr. Rosenblum for a print from which Fig. 10 was made.

'^ This curve was the first to disclose the a-rays of RaC, previously known only
by inference (though it was very compelling inference). Being of somewhat lesser
range than the much more numerous (to be precise, 3000 times as numerous) a-
particles emanating from the RaC atoms with which RaC is always inevitably
mingled, they were completely hidden from observation by any method known
before the use of the differential chamber and the powerful magnet.

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 589

at least five cases in which the distribution-in-range curve obtained
with the differential chamber shows a single symmetrical peak sug-
gesting only one group: from the radium series, Rn and RaA; from
the thorium series, Tn and ThA; from the actinium series, AcA.
Altogether there are twenty-three ^^ known alpha-emitters, so that
nearly half of the total remain to be investigated to this end. It may
be significant that out of the four known alpha-emitters having odd
atomic number, the high proportion of three at least is known to
display fine-structure (the fourth, Pa, being as yet uninvestigated).

Interrelations of Alpha-Ray Spectra and Gamma-Ray Spectra

Evidently, if two atoms of the same radioactive substance were to
emit alpha-particles of different speeds, there would be three obvious
possibilities. The resultant nuclei might be different: in this case
we should expect (though not with certainty) that they would be the
starting-points of different radioactive series, and we should speak of
"branching." The initial nuclei might have been different, in which
case it would have been improper to speak of them as belonging to the
same substance. Finally one at least of the two atoms might also
emit gamma-ray photons, of energies complementary to those of the
alpha-particles, in such a way that the total amount of energy released
by the one atom would be the same as the total amount released by
the other.

The first of these possibilities is not to be excluded a priori (since
there are known cases of branching, though in them the alternative
is between emission of an alpha-particle and emission of an electron)
and neither is the second. The third, however, is the most agreeable,
since if realized it allows us to believe that in the transformation of
radium (to take one example) every radium nucleus is like every other
before its change begins and every resulting (radon) nucleus is like
every other after its change is over. Now alpha-ray emission and
gamma-ray emission often occur together, which suggests that often
the third possibility is the one which is realized; but this cannot be
proved without measuring the energies or the wave-lengths of the
gamma-rays.

The simplest cases are those in which the alpha-ray spectrum con-
sists of two lines only. Here and always, there is an inconvenient
complication- at the start: when an alpha-particle is emitted, the
residual nucleus recoils, and it is the sum of the kinetic energies of

'^ Not including Sm and other elements of atomic number lower than 81. The
rest are depicted (together with the beta-ray emitters of atomic numbers 81 and
greater) in Fig. 21.

590 BELL SYSTEM TECHNICAL JOURNAL

the two (not that of the alpha-particle alone !) which must be taken
into account.^^ Denote by C/i and U^ the values of this sum for the
faster and for the slower alpha-particles. Does the gamma-ray
spectrum then consist of a single line of which the photon-energy hv is
equal to ( t/i - f/2) ?

In the case of AcC, the difference {Ui - U2) is 0.35 or 0.36 MEV.
There is an intense gamma-ray line proceeding from actinium active
deposit (comprising AcC), and the energy of its photons is concordant.
In the case of RaC, the difference {Ui — U2) is only 0.04 MEV, and
the search for so relatively soft a radiation of photons is difficult.
In the case of Ra the conditions are more favorable, and here the
history is worth retelling. Long before the earliest analysis of alpha-
ray spectra, radium was known to emit feeble gamma-rays of photon-
energy about 0.19 MEV. An estimate of their intensity was made in
1932 by Stahel; he concluded that the photons are less than one-
tenth as numerous as the alpha-particles already known. Search was
thereupon made by Rosenblum for fine-structure in the alpha-ray
spectrum of radium. Two lines appeared on the plate after five
minutes' exposure: they were due to groups proceeding one from
radium and the other from its daughter-element radon. On plates
exposed for hours there appeared yet another line. The values of
Ui and U2 being computed for this and for the stronger radium group,
the difference was found to be close to 0.185 MEV, with a sufficient
latitude to be concordant with the estimate for the photons.

As the number of alpha-ray lines increases beyond two, the prospects
rapidly become formidable; for a spectrum of n such lines suggests n
possible states of the residual nucleus, and every one of these might
"combine" (in the technical sense of the word) with every one below
it in the energy-scale, making a total of n{n — l)/2 gamma-ray lines
to be expected. Even so, anyone acquainted only with optical spectra
might think it no difficult matter to photograph (say) the gamma-ray
spectrum of ThC, and see whether it consists in just 15 lines in just
the right places to correspond with the six alpha-particle groups.
But one does not photograph gamma-ray spectra — one photographs
the beta-ray spectra of the electrons ejected by the gamma-rays from
atoms, and tries to deduce the photon-energies hv from the electron-
energies.^^ The atoms may be those of the radioactive substance

" By multiplying the kinetic energy of the alpha-particle by the factor (1 -|- m/M),
where m stands for the mass of the alpha-particle and M for that of the recoiling
nucleus. This point was overlooked by a number of people before it was noticed by
Feather.

1* I have dealt with this procedure at length in the article "Radioactivity," No.
XII of this series (Bell Sys. Tech. Jour., 6, 55-99, 1927).

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 591

itself (either the very ones which are emitting the gamma-rays, in
which case the rays are said to undergo "internal conversion," or
their neighbours) or they may be those of other elements mixed with
the radioactive substances, or those of nearby solids or gases on which
the gamma-rays fall. Each gamma-ray line is responsible for several
different beta-ray lines, a circumstance which makes the analysis
more difficult at the beginning though it makes the inference more
reliable in the end. There may be gamma-rays having nothing to do
with alpha-particle emission, and there may be gamma-rays from
several different radioactive substances inextricably mixed up together,
so that the problem of analyzing the spectrum of one transformation is
preceded (or, more truly, accompanied) by that of distinguishing it
from the intermingled spectra of others. ^^ The experimental errors in
the estimates of L'^- values and /zi'-values may be so large that apparent
agreements are actually unreliable. Altogether, the comparison of a
rich alpha-ray spectrum with a rich gamma-ray spectrum is an
exceedingly intricate business, the outcome of which is not to be
summarized in a few sentences. To give a mere notion of the sort of
conclusion which is reached, I quote some lines from Rutherford,
Lewis and Bowden, in their comparison of the thirteen-line alpha-ray
spectrum and the very rich gamma-ray spectrum of RaC:

"When we consider in broad terms the data which have been
presented, there can be no doubt that there is a high correlation
between the alpha-particle levels which have been observed and the
emission of gamma-rays. In more important cases the numerical
agreement is well within the experimental error of measurement,
while the relation between the intensity of the alpha-ray groups and
the gamma-rays associated with them is of the right order of magnitude
to be expected on general theoretical grounds. In other cases the
agreement is very uncertain, and more definite information on the
gamma-rays is required to make the deductions trustworthy. It is
unfortunate that we have been unable to detect the alpha-particle
groups corresponding to certain postulated levels [i.e. postulated from
the classification of the gamma-ray lines] . . . ."

Thus it appears that there are excellent agreements between hv-
values and {Ui — Uj) values, and yet nothing approaching a perfect
one-to-one-correspondence. Nevertheless, the general programme is
fixed: to assume that each nucleus possesses a system of stationary

" It is interesting to notice that after the /zj/- values of certain gamma-rays emitted
from mixtures of ThC and ThC" had been found to agree with values of {Ui — Uj)
taken from the alpha-ray spectrum of ThC, these gamma-rays were proved to proceed
from ThC in its transformation into ThC", whereas till then they had been supposed
to proceed from ThC" in its transformation into ThD (Meilner & Philipp, Ellis).

592 BELL SYSTEM TECHNICAL JOURNAL

states and energy-levels, to assume further that hv-vaXnes and
{Ui — C/y)-values are alike the differences between these energy-levels,
and to ascribe apparent defects of correlation to special circumstances
by virtue of which certain gamma-rays and certain alpha-rays are
too feeble to be detected. Should these ideas prove untenable, we
shall probably have to suppose that the nucleus is even more different
from the extra-nuclear world than we have hitherto admitted.

Now arises the important question: when alpha-particles and
photons both are emitted in the course of the complete transformation
of one nucleus into another, which comes first? Despite the im-
measurable shortness of the times which are involved, this is in
principle an answerable question. For as I have mentioned already,
gamma-rays are detected and their photon-energies are measured by
examining the spectrum of the electrons which they eject from the
orbital electron-layers of atoms, chiefly from the layers surrounding
those very nuclei whence the photons themselves proceed. Now if
the photons come before the alpha-particles, say for example in the
transformation of ThC into ThC", these electrons will come from the
electron -layers of ThC atoms; in the contrary case, from the layers
of ThC". It is possible to distinguish from which they do come,
even when the energy of the photons is not independently known and
must itself be derived from the same data.^"

The classical and crucial experiment of this type was performed
about ten years ago by Meitner, and it proved that the gamma-rays
emitted during the transformation of RdAc into AcX and during that
of AcX into An spring forth after the alpha-particle has departed and
the nucleus has become that of the daughter-element. These are
two of the cases in which the alpha-ray spectrum exhibits fine-
structure; and it is now generally supposed that the rule extends to all
such cases. The stationary states or energy-levels deduced from the
{Ui — f/y)-values and the /zj^-values then would pertain to the "final"
or daughter nucleus. In the instances where all the alpha-particle
groups except the main one are designated "long-range groups" —
RaC and ThC (the quotation above from Rutherford, Lewis and
Bowden refers to the former of these) — Gamow argues that the gamma-
rays are emitted before the alpha-particles; the energy-levels deduced
from the alpha-ray and the gamma-ray spectra would then pertain to

^^ See the previously-cited article "Radioactivity," pp. 94-96. I mention in
passing that sometimes the "internal conversion" of photons whereby electrons are
ejected is apparently so much the rule, that no appreciable fraction of the gamma-
rays of some particular energy (or energies) escape from the atoms at all; in which
cases it becomes expedient to speak not of gamma-rays at all, but of an immediate
transfer of energy from the nucleus to the orbital electrons (a policy which may be
applied to all cases of internal conversion).

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 593

the "initial" or mother nucleus. It is not clear from the litera-
ture whether this hypothesis has been fully tested in the manner of
Meitner's tests aforesaid, but presumably it was adopted in calculating
the ^j^-values from the electron-energies, so that the agreements
between hv and ( Ui — Uj) support it.^^

The search for interrelations among the energy-levels, the different
hp-values and the different f/-values belonging to individual trans-
formations has of course already begun. Rutherford and Ellis find
that the frequencies of many of the lines in the gamma-ray spectrum
of RaC can be fitted by assigning various integer values to p and q and
constant values to £i and £2 in the formula pEi -\- qE^; while H. A.
Wilson finds that if the tZ-values or the /zj'-values are added together
in pairs, an amazing number of the pairs are equal to integer multiples
(the integer multipliers ranging from 16 to 54) of the amount 0.385
MEV — this even if the two members of a pair are taken from different
spectra !

The Quantum-Mechanical Theory and the Crater
Model of the Nucleus ^^
Anyone who is acquainted with the contemporary atom-model in
its present or in its earlier stages, with its congeries of charged particles
revolving in or jumping between definitely-prescribed and quantized
orbits, governed by attractions and repulsions both classical and
unimaginab' 2 — any such person will probably be looking for a nucleus-
model of the same variety but built on a very much smaller scale,

" I learn by letter from Dr. Ellis that in the case of RaC, some at least of the
gamma-rays which agree with the ([/,-C/,)-values of the long-range alpha-particles
are definitely proved in this fashion to proceed from nuclei of atomic number 84
(that of RaC') as distinguished from 83, 82 or 81; the proof is especially strong for
the most intense gamma-ray, of photon-energy 0.607 MEV. Perhaps this is the most
powerful evidence that the long-range particles come from RaC rather than RaC.

The half-periods of RaC and ThC' are exceedingly short, 10~^ sec. and 10~^^ sec.
respectively; had it been otherwise, objection might be made to Gamow's contention
on the ground that atoms in states, from which they are liable to depart by emitting
radiation, generally do depart from those states and emit that radiation within a
period of the order of 10~^ or 10"* sec. There is no a priori certainty that this
principle applies to nuclei, but if it does it may suffice to explain why the long-range
particles are observed only from these very short-lived nuclei, why they are so
scanty even in these cases (nearly always the photon is emitted before the alpha-
particle gets ready to leave, so that the latter nearly always leaves with low energy
instead of high), and why the fine-structure of other alpha-ray spectra is related to
energy-levels of the final instead of the initial nucleus (Gamow). Incidentally it
strengthens the case for ascribing the long-range particles to the C-products insterd
of the C-products.

^2 Quantum mechanics was first applied to the nucleus-model here to be described,
independently and almost simultaneously, by Gurney and Condon and by Gamow.
Rather than by all three names together, it seems preferable to denote this model
thus interpreted by a neutral descriptive term, such as "crater model" (an allusion
to the aspect of the graph obtained when the curve of Fig. 12 is coupled with its
mirror-image in the yz-plane).

594

BELL SYSTEM TECHNICAL JOURNAL

with protons and possibly neutrons figuring among the revolving
particles. He will be looking too far into the future, and will be dis-
appointed with the present. The present nucleus-model consists of
little more than a single curve — a curve which, moreover, relates
only to the fringe of the nucleus and to the region surrounding it, and
for want of knowledge is not extended into the central region or
nucleus proper where the constituent particles must be. The theory
which it serves is a theory not of the nucleus as a stable system of
corpuscles, but of the escape of some from among these corpuscles
and the entry of new ones — a theory professing to deal only with the
entry and the escape, not at all with the events succeeding the one or
preceding the other.

The curve purports to portray the electrostatic potential, as function
of r the distance measured from the centre of the nucleus, from r = co
inward to a minimum distance which is indeed very small even in the
atomic scale — 10"^^ cm. or less — but still definitely not zero, since the
components of the nucleus must be presumed to be normally at
distances yet smaller. When it is plotted as in Fig. 12, its ordinate

Fig. 12 — Nuclear potential-curve postulated for explaining transmutation (without
allowance for resonance) and radioactivity.

at any r is a measure of the amount of kinetic energy which a posi-
tively-charged particle approaching the nucleus must sacrifice — i.e.
which must be converted into potential energy — in order to come
from infinity to r. Traced from infinity inward, the curve must
follow at first the function const, jr, corresponding to the inverse-
square law of force; for it is known, both from experiments on alpha-
particle scattering (which supplied the foundation for the contemporary
atom-model) and from the successes of the theory of atomic spectra,
that beyond a certain distance a nucleus is surrounded by an inverse-

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 595

square force-field. This is the obstacle, or at any rate, a part of the
obstacle, which an oncoming proton or deuton or alpha-particle must
overcome in order to reach the nucleus and achieve transmutation.
One may picture it as a hill, up which the ball must roll to reach the
castle at the top — and down which the ball will roll if it starts from
the top, shooting outward towards infinity as the fast-flying alpha-
particle.

Now to assume the inverse-square force as prevailing all the way
inward to r — 0 would be to postulate a point-nucleus without room
for parts or structure, surrounded by a hill of infinite height which no
approaching positive particle could climb; all of which is inadmissible.
Departure from the inverse-square law is actually shown by some
experiments on scattering of alpha-particles which pass very close
to the nucleus, and these indications are to be heeded in tracing that
part of the curve of Fig. 12 which lies to the right of the maximum;
but the maximum itself and the sharp descent to its left are dictated
by no such observations, and to postulate them is to make the theory
which is now to find its employment and its test. That there should
be such a maximum and such a descent is of course the most natural
supposition to make. If there are several particles of positive charge
which stay for a finite time within the nucleus, there must be something
which restrains them from flying away. This something must either
be an agency of a type as yet unknown, or else be described by a
potential-curve with a maximum at what we may henceforward call
the boundary of the nucleus; and the latter assumption is to be pre-
ferred till proved unusable.

Applying classical ideas to this "model" (if the word be not con-
sidered too presumptuous) of a nucleus, one is led at once to two pre-
dictions, which may be sharply formulated if we adopt symbols such
as Vm and r™ for the two parameters indicated on Fig. 12, viz. the
"height of the potential-barrier" and the "radius of the potential-
barrier" as they are commonly called, the latter being also called the
"radius of the nucleus." These are:

1. If the nucleus emits a particle of positive charge -f 2e, the
kinetic energy with which this particle is endowed when it completes
its escape cannot be less than 2eVm', consequently, when it is observed
that atoms of a certain element emit alpha-particles with kinetic
energy Ko, the height of the potential-barrier for that element cannot
surpass Ko/2e; consequently, when the force-field about the nuclei of
such atoms is explored by the classical method of studying the scatter-
ing of alpha-particles projected against a sheet of that element, it
must be found that the region of repulsive force, and a fortiori the

596 BELL SYSTEM TECHNICAL JOURNAL

inverse-square field, do not extend far enough inward for the integral to
surpass the value Ko/le.

2. When particles of charge ne {n = any integer) are projected at
a sheet of any specific element, they cannot enter the nuclei at all
unless their kinetic energy exceeds the critical value neVm', and if the
curve of number-entering-nuclei vs. kinetic-energy can in any way be
deduced from any experiments, it should rise fairly sharply from the
axis of kinetic energy at this critical abscissa.

(It will have been noticed that I expressed both of these predictions
as though the escape or the entry of a particle made no difference to
the height of the potential-barrier, which is the universal practice.
This is obviously too crude an assumption; the error in it must be
graver the smaller the atomic number of the element, therefore graver
in theorizing about the transmutation of light elements than in
theorizing about the radioactivity of heavy ones; it must be rectified in
future.)

The former of these predictions can be sharply and unquestionably
tested; and it proves to be wrong. Uranium I. emits alpha-particles of
kinetic energy Kq equal to 4 MEV; but Rutherford suspected from
scattering-experiments on other heavy elements, and subsequently
proved by such experiments upon uranium itself, that the inverse-
square force-field extends so far inwards as to involve a height of
potential barrier at least twice as great as Ko/2e; so that an emerging
alpha-particle should possess at least 8 MEV of kinetic energy derived
from coasting down the hill, and even this is merely a lower limit to
the estimate, since the hill may be higher and the particle might
come with some excess of energy over its brow!

The second prediction is not so readily tested. If all of the charged
particles (protons or deutons or alpha-particles, say) projected at the
postulated sheet of matter were directed straight towards the centres
of nuclei, and arrived at the potential-hills without suffering any
prior loss of energy elsewhere, the fraction entering through the
potential barriers would rise suddenly from zero to unity as the kinetic
energy K of the particles was raised to neVm, and any phenomenon
depending solely upon entry would make its advent suddenly if at all.
Unfortunately this does not occur in any experiment now possible or
likely ever to become possible. If the sheet of matter is a monatomic
layer, most of the oncoming particles will be going towards the gaps
between the nuclei, and the initial directions of the rest will be pointed
towards all parts of the cross-sections of the nuclei, only an infinitesimal
fraction going straight toward the centres. Designate by p the
perpendicular distance from a centre to the line-of -initial-motion of

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 597

an oncoming particle; it is evident that the minimum kinetic energy
permitting of entry will increase with p, starting from neVrr, and rising
to infinity as p rises from 0 to r™. The relation between fraction-of-
particles-entering-nuclei — call it Pe — and kinetic energy K could be
calculated, given specific assumptions about the values of F„ and r„,
and the trend of the potential-curve. Without undertaking the calcu-
lation, it is easy to see that the vertical rise of what I will hereafter
call the "ideal" curve — the curve of probability-of-entry-at-central-
impact vs. K — will be distorted into a bending slope, starting, however,
at the same critical abscissa neVm. If the sheet of matter is a thick
layer, there will of course be a much greater fraction of the impinging
particles of which the initial paths point straight toward some nucleus
or other, but the fraction achieving entry will not be raised in the same
ratio, for the particles going toward nuclei embedded deep in the
layer will lose some or the whole of their velocity in passing through
the intervening matter.-^ This also will contribute to converting the
vertical rise into a gradual bend. Still it does not seem possible that
if the ideal curve had such a shape, the experimental ones could rise
with so extreme a gradualness as does the one of Fig. 17 or those of
Figs. 16 and 17 in the Second Part; for these suggest no sudden be-
ginning at all, but rather they have the characteristic aspect of curves
asymptotic to the axis of abscissas, as if their apparent starting-points
could be pushed indefinitely closer to the origin by pushing up indefi-
nitely the sensitiveness of the apparatus. Neither does it seem possible
that Vm can be so low as their starting-points imply.

There is, however, another difficulty: these curves refer not directly
to Pe, but to number of transmutations, or to be precise (for precision
is essential in these matters) to the number of particles producing
transmutations involving the ejection of fragments having certain
ranges. Call this number Pt. It is easiest to conduct the argument
as though Pt were proportional to Pe — as if an observable transmuta-
tion could result only from the entry of a particle through the potential-
barrier of a nucleus, and as if the number of transmutations of any
special type were strictly proportional to the number of entries, the
factor of proportionaHty being independent of K. Yet few assump-
tions are less plausible. It is far more reasonable to suppose that the
probability of a particle bringing about a transmutation when it enters
a nucleus is not invariably unity, but is instead some function /i(J^).
It is reasonable also to suppose that a particle passing close to the
potential-barrier but not traversing it may yet be able to touch off an
internal explosion or eruption leading to a transmutation. Denote

2^ Contrast the two curves of Fig, 17.

598 BELL SYSTEM TECHNICAL JOURNAL

by fi{K){\ — Pe{K)) the number of cases in which this happens.
The least which we can take for granted is some general relation of
the form,

Pt^MK)-P.iK) -^MK){1 - P.{K)), (20)

and the variations of /i and f^ may contribute still further to blotting
out all signs of the hypothetical vertical rise in the ideal curve. More-
over,/2 might be appreciable at values of K smaller than neVm, thus
blotting out every sign of the critical energy-value at which entry
commences.

Thus with regard to the second prediction, the situation is this:
the experimental curves of number-of-observed-transmutations vs.
kinetic-energy-of-impinging-particles rise so smoothly and so gradually
from the axis as to give not the slightest support to the idea that
entry into the nucleus commences suddenly at a critical value of K;
moreover, transmutation commences to be appreciable — -for several
elements, at least — when K is still so small that K/ne is only a small
fraction of the least value which can reasonably ^* be ascribed to Vm,
in view of what we know from alpha-particle-scattering about the
circumnuclear fields of these or similar elements. This again might be
due to the hypothetical effect to which the term fiiK) in the equation
alludes, but it seems far too prominent for that! With it is to be
linked the fact that alpha-particles emerge from nuclei with kinetic
energy less than 2eVm. The potential-hill seems not to be so high
either for entering or for emerging particles, as it is for those which
only ^kirt its slopes !

Now if in theorizing about potential-hills and particles we substitute
quantum mechanics for classical mechanics, these phenomena cease
to be things contrary to expectation, and become instead the very
things to be expected.

This is one of the situations — regrettably frequent in the present-day

theoretical physics — where neither pictures nor words are adequate.

The nearest description which can be made with words is probably

somewhat as follows: We set out to ascertain whether a particle of

charge ne and kinetic energy K, coming from infinity straight toward

the nucleus (I simplify the problem as much as possible) will surmount

the potential-hill of height Vm- Were we to conceive it as a particle

conforming to classical mechanics, we should arrive at the answers:

yes, ii K = neVm — no, \l K < neVm- But we are to turn away from

^^ It is true that the elements of which the circumnuclear fields have been most
carefully explored by alpha-particles are not in general the same as those for which
transmutability has been observed down to very low values of K; but boron and
carbon figure on both the lists, Riezler having studied the scattering of alpha-
particles by these {Proc. Roy. Soc, 134, 154-170, 1932).

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 599

the particle for awhile, and to conceive a train of waves advancing
from infinity towards the nucleus. The phase-speed and the fre-
quency of this wave-train are prescribed by definite rules making them
dependent upon E, and the train is governed by a prescribed wave-
equation in which figures the function Vir) of Fig. 12. On solving
this equation in the prescribed fashion we find that it requires the wave-
train to continue (though reduced in amplitude) past the top of the
hill if K is greater than neVm. This is partially satisfactory, for the
particle when it is reintroduced is to be associated with the waves,
and everything would be spoiled if the particle could go where the
waves cannot. But also, the equation requires the wave-train to
continue past the top of the hill when K is less than neVm- True, it
does not wholly pass; there is a reflected as well as a transmitted beam,
and the ratio of reflected to incident amplitude goes very rapidly up
towards unity and the ratio of transmitted to incident amplitude goes
very rapidly down toward zero as K drops downward from the value
neVm- All the same there is this wave-train beyond the hill with an
amplitude greater than zero; and the association of particles with
waves is apparently spoiled, for the waves can go where the particle
cannot.

At this point, however, it is the rule of theoretical physicists to
give the precedence to the waves, and declare that where the waves go
there the particle must go also, whether it can (by classical mechanics)
or cannot. Since some of the waves are beyond the hill, the particle
also must be able to traverse the hill, even though its kinetic energy is
insufificient for it to climb to the top. But since the waves beyond the
hill have a smaller amplitude than those coming up from infinity, it
is not certain that the particle will pass through, but merely possible.
The chance or probability of its passing through is determined chiefly
(not fully) by the ratio of the squared-amplitudes of the waves on the
two sides of the hill, and this is what must be computed by quantum-
mechanics. How the particle gets over or through the hill — -where
and what it is and how it is moving while it is getting through — these
are questions which the theorist usually declares to be unanswerable
in principle, and having so declared, he does not attempt to visualize
this part of the process.

Into Fig. 12 the diagonal lines have been introduced in a crude
attempt to make graphic as much as possible of the theory. The
length of the sloping line drawn from any point P of the curve is
meant as a sort of inverse suggestion of the chance which a particle
of charge -f ne has of entering the nucleus if its energy E is equal to
ne times the ordinate of P: the longer the line, the less the chance of

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BELL SYSTEM TECHNICAL JOURNAL

entry! (This energy E will be the same as the initial kinetic energy
K already so often mentioned, which the particle has before it starts
to climb the hill.) Perhaps it is not too fanciful to think of these
lines as the posts of a fence standing up vertically from the curve, the
varying height of which is a rough indication of the varying difftculties
which particles of various energies have in getting through.

To predict successfully how the height of this metaphorical fence,
the probability of transmission or of penetration, varies with V or E
would be a magnificent triumph of nuclear theory, but it is vain to
hope for such a success in the immediate future. Almost certainly the
top of the fence curves much more rapidly upward than the drawing
suggests, and also there is good reason to think that there may be gaps
in the fence somewhere like those depicted in Fig. 13, where the

Fig.

13 — Nuclear potential-curve postulated for explaining transmutation, with
allowance for " resonance."

probability of penetration rises to values remarkably near to unity.
Such at least are the features of certain one-dimensional potential-
fields (do not forget that Figs. 12 and 13 refer to three-dimensional
potential-fields having spherical symmetry!) which have isolated
potential-hills or hills-adjoined-by-valleys.

Three of these cases are displayed in Figs. 14, 15, 16. Take the first
for definiteness. One plunges in medias res by writing down at once
Schroedinger's wave-equation :

(p-^ldx- + (87r2m//z2)[£ - neV{x)']^ = 0,

(21)

in which V{x) stands for the potential-function exhibited in the
figure,25 while the meanings of ne, m and E have probably already been

2^ I deviate from the otherwise-universal usage of employing V for the potential
energy of the particle, for the reason that the latter depends on the charge of the
particle, while the potential-function is supposed (no doubt inaccurately) not to
depend on it.

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII

601

-F

_i_

E/ne

Fig. 14 — Illustrating an artificial case of a potential-curve with' a single square-
topped potential-hill.

Vm E/ne

Fig. 15 — Illustrating an artificial case of a potential-curve with a pointed hill.

E/ne

Fig. 16 — Illustrating an artificial case of a potential-curve with a valley between

two hills.

guessed by the reader — they are constants to which any values may
be assigned, and the eventual result of the mathematical operations is
going to be taken as referring to a stream of particles of charge ne,
mass m and energy E.

The problem is stated as that of finding a solution of (21) for what-
ever value is chosen for E — a solution everywhere single-valued,
bounded, continuous, and possessed of a continuous first derivative,
such being the general requirement in quantum mechanics. Not,
however, any solution possessing these qualities, but a solution apt to
the physical situation. On the right of the hill, it must specify a
wave-train (I) going from right to left; for we are interested in the
adventures of particles coming from the right toward the hill. But
on the right of the hill, it must also be capable of specifying a wave-
train (II) going from left to right, for some or all of the particles may
be reflected from the hillside. On the left of the hill it must be capable
of specifying a wave-train (III) going from right to left, for some or
all of the particles may traverse the hill and continue on their way.

602 BELL SYSTEM TECHNICAL JOURNAL

So far as the region to the right of the hill {x > Xa) is concerned, a
solution having all of these qualities is the following:

<l, = A le-^^-V^ + A^e+'^-<^, k = J^^ ,

(22)

in which the two terms stand for wave-trains I and II, and Ai and A2
are adjustable constants. So far as the region to the left of the hill
(x < Xi) is concerned, a solution having all of the required qualities is
the following:

^ = Cie-^'^^'^. (23)

It stands for wave-train III and Ci is an adjustable constant. As I
have already said, our "intuition" based on notions of what particles
should do, expects Ci to vanish and ^2 to become equal to Ai when E
is less than neVm, but the solution of (21) does not consent to these
limitations. Our intuition also expects that when E is less than neVm
nothing will happen in the region comprised within the hill (xi<x<X2),
but here again the solution of (21) does not conform with it. For
in this region comprised within the hill, the solution must take the form :

^ = 5^g-^x^„.r,„-£ ^ 526+^^^^"^^'"-^ (24)

which looks at first glance like (22) but is essentially different, since
the exponents are real and not imaginary, and the terms do not repre-
sent progressive waves. The five coefficients — Ai, A2, Bi, B2, Ci —
must now be mutually adjusted so that at the sides of the hill (x = Xi
and X — X2) the expressions (22) and (23) and (24) flow smoothly each
into the next, with no discontinuity either of ordinate or of slope.
This imposes four conditions on the five coefficients, and therefore
fixes the relative values of all of them — in other words, determines
them completely except for a common arbitrary factor which corre-
sponds to the intensity of the incident beam, and is irrelevant to the
course of the argument.

In particular, this requirement of continuity imposed by the funda-
mental principles of quantum mechanics upon the acceptable solution
of (21) fixes the ratio of the amplitudes d and Ai of "transmitted"
and "incident" wave-train. From Gurney and Condon I quote an
approximate formula -^ for the ratio of the squares of these amplitudes,
denoting them on the left by the customary symbols:

^7^2T^' = ^^^^ ^'^P- ^- (4^«//0V2m(iVeF,„-£)],

V^^ Jinc. i^2S)

4>{E) = \6{ElneV^){\ - E/neVm).
^^ Exact formula given by E. U. Condon, Reviews of Modern Physics, 3, 57 (1931).

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 603

This expression does not vanish suddenly as soon as E drops below
neVm, but falls away continuously — and very rapidly, it must be
admitted, owing to the exponential factor — as E diminishes from neVm
on downwards. Its value depends on a, the breadth of the hill (Fig.
14) in such a way that the broader or thicker the hill of given height
the less the amplitude of the transmitted waves: the thicker the hill,
the more nearly it comes to fulfilling the classical quality of being a
perfect obstacle to particles having insufficient energy to climb it!
I rewrite (25) in the equivalent form,

(^^*)t,

(^^*)ine.

- (47r//o r

= 4>{E) exp. [- (47r//0 ■ ^2m{NeVm - E)dx, (26)

the integral in the exponent being taken "through the hill" from Xi
to X2. This form is generalizable. Take the case of Fig. 14: the ratio
of the squared-amplitudes of transmitted and incident wave-trains is
given, according to Fowler and Nordheim, by an expression which is
of the type (26), except that 0(-E) is a somewhat different function
(it is A:i{EINeVm)0~ - ElNeVm)Ji-). The distance from Xi to X2,
over which the integration is carried, obviously depends on E in this
and every other case but the particular one of Fig. 14. Take finally
the general case of a rounded hump, such as appears in Fig. 12.
According to Gamow, a formula of type (26) is approximately — not
exactly — valid for every such case, ^(.E) being given by him as
simply the number 4 when the hill descends to the same level on
both sides as in Fig. 14; while in the general case where the potential-
curve approaches different asymptotes at — co and + oo — say zero
at the latter, Vr ait the former — the factor 4>{E) assumes the form
A[EI{E — neVr)'Ji''-. Now E was the kinetic energy of the particles
at infinity in the direction whence they come, and {E — neVr) will
be the kinetic energy of the particles at infinity in the direction whither
they are going. We have been denoting the first of these quantities by
K\ denote the second by Kr, and the corresponding velocities by v and
Vt. Then 0(£) can be written as ^vjvr.

The question must now be answered: what is the actual relation
between the ratio (^^*)trans./(^^*)inc., and the probability that a
particle will traverse the hill? In associating waves with corpuscles,
it is the rule to postulate that the square of the amplitude of the
waves at any point is proportional to the number-per-unit-volume
of corpuscles in the vicinity of that point. If one prefers to think of a
single particle instead of a great multitude, one may say that the square
of the amplitude of the waves at any point is proportional to the proba-

604 BELL SYSTEM TECHNICAL JOURNAL

bility of the particle being at that point. Let us hold, however, to the
picture of a dense stream of corpuscles approaching the potential-hill —
say that of Fig. 14 — from the right, and a much weaker stream receding
from it on the left. If there are 5 times as many corpuscles per-unit-
volume on the right as on the left, then there cannot be a steady flow
unless only one out of 5 incident particles traverses the hill. The re-
ciprocal of 5 is the fraction of particles getting through the hill, or the
probability of a single particle getting through; and it is also the
ratio ("^^*)trans./("^^*)inc.. This statement, however, is too narrow,
being valid for the case where the speeds v and Vr of the particles
on the two sides of the hill are the same. In the general case, we have :

Probability of transmission or penetration

= (V^)(^**)trans./(^^*)inc.

= {vrlv)-4>{E)-exp.l- {lirlh) p dx^2m{E- neVm)^. (27)

In Gamow's approximation the product of the first two factors has the
pleasantly simple constant value of 4. In the approximations of
Gurney and Condon and of Fowler and Nordheim for the cases of
Figs. 14 and 15, the product is some function of E which the reader
can construct from the foregoing equations. In all these cases, how-
ever, it is the exponential factor which dominates the trend of either
member of (27) considered as function of E.

Now immediately one sees, that if transmutation is due to the
penetration of a charged particle through a potential-hill or potential-
barrier surrounding a nucleus — and if this penetration is governed by
laws of quantum mechanics as illustrated in the one-dimensional cases
— then when the number of observed transmutations is plotted against
the kinetic energy K of the impinging particles, the curve should be
expected to rise with a gradual smooth upward curvature from the
axis of K', and there should be no critical minimum value of K for the
advent of the phenomenon, but rather the beginning of perceptible
transmutation should be observed at progressively lower and lower
energy-values, as the sensitiveness of the detecting-apparatus is im-
proved ; and it may well be that transmutation can be detected when
K is still so low, that the quotient of K by the charge of the particle
is far smaller than any reasonable guess that can be made of the
height of the barrier. All these are features of such curves as those
of Fig. 17, or Figs. 16 and 17 of the Second Part. The adoption of
quantum mechanics permits us to accept these features without ascrib-
ing them to the hypothetical functions denoted by/i and/2 in equation

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII

605

2200

a 1200

1000

400

200

1
1

1
1
1

1
1

1
THICK /
LAYER ;

;

LOG /

1 /
/ /
; /

/ / THIN
1 / FILM

/
/
1

1 1
1/
1/

3.5

3.0

0.02 0.04

0.08 0.10
MFV

0.12

Fig. 17 — Transmutation of boron by impact of protons: rate of observed trans-
mutation as function of K, for a very thin film and for a thick layer. (Oliphant &
Rutherford, Proc. Roy. Soc).

(20), though it does not rule out the possibility that these functions
may have influence upon the curves.

But not all of the curves of probability-of-transmutation versus K
are of the simple type of Fig. 17. There are also some which show
distinctly-marked peaks superimposed upon the gradual upward sweep ;
that of Fig. 18 for example, which relates to the transmutation of
beryllium by impact of alpha-particles with emission of neutrons,
presumably by the process

2He4 + 4Be» = eC^^ + on^

and that of Fig. 19, which relates to the presumptive process,

2He4 + ,^^AF = ^,^si^» + ,UK

606

BELL SYSTEM TECHNICAL JOURNAL

■* 200

P 150

10 2.0 30

RANGE OF INCIDENT CC-PARTICLES

Fig. 18 — Transmutation of beryllium by impact of alpha-particles, with produc-
tion of neutrons; rate of observed transmutation as function of K, for a very thin
film and for a thick layer (dashed and full curves respectively), illustrating resonance.
(Chadwick; Proc. Roy. Soc).

3.6 3.5 3.4 3.3

MAXIMUM RANGE OF OrPARTICLES

Fig. 19 — Transmutation of aluminium by impact of alpha-particles, with produc-
tion of protons; rate of observed transmutation as function of residual range of alpha-
particles, illustrating resonance. (Chadwick & Constable; Proc. Roy. Soc).

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 607

In Fig. 19 the abscissa is not K, but a quantity (the range of the
impinging alpha-particles) which increases more rapidly than K\ but
this does not affect the meaning of the peaks. Moreover, there is
abundant indication that quantities of such curves are simply waiting
for someone to take the data and plot them; for this is the phenomenon
of "resonance" to which many pages ^^ were devoted in the Second
Part, and which has chiefly been observed by the other methods there
described, but should always manifest itself in this way when the
proper experiments are performed.

If we wish to interpret this without letting go of the classical theory,
we must say that either or both of the functions /i and/2 have maxima
at certain values of K. But here again, the adoption of quantum
mechanics may make this step superfluous. For consider the one-
dimensional potential-distribution of Fig. 16, a valley between two
hills, with energy-values reckoned from the bottom of the valley.
If the wave-equation be solved for this potential-distribution and for
any such value of particle-energy E as the dashed line of. Fig. 16
indicates — such a value, that according to classical theory a particle
possessing it might either be always within the valley or always beyond
either hill, but never could pass from one of these three zones to
another — a curious result is found. For the solutions which the laws
of quantum mechanics demand and accept, the ratio of squared-
amplitude ^^* within the valley to squared-amplitude ^^* beyond
either hill is usually low, but for certain discrete values of E it attains
high maxima!

Now the three-dimensional nucleus-model of which I am speaking
resembles this case more than it does the other one-dimensional cases
of Figs. 14 and 15, because it consists of a potential-valley surrounded
on all sides by a potential-hill. One may therefore expect the prob-
ability of entry or penetration to pass through maxima such as are
symbolized by the dips in the "fence" of Fig. 13, entaiUng maxima
in the curve of probability-of-transmutation P« plotted as function
of K. Such is the quantum-mechanical explanation of the phe-
nomenon of "resonance," which indeed derives its name from this
theory; for the values of X" or £ at which the maxima occur are those
for which the amplitude of the oscillations of the ^P-function in the
valley within the barrier are singularly great.

One wants next to know what quantitative successes have been
achieved in predicting or explaining such things as the actual locations
of the resonance-maxima, or the precise trend of the curve of Prvs-X

27 "Nucleus, Part II," pp. 148-153; more fully treated in Rev. Set. Inst., 5, 66-77
(Feb. 1934).

608

BELL SYSTEM TECHNICAL JOURNAL

as it rises away from the axis of abscissse. Here it must be admitted
that almost everything remains to be done. The locations of the
resonance-maxima must be expected to depend upon the details of the
potential-distribution within the valley, of which there is as yet no
notion. The precise trend of Pi as actually observed cannot be the
same as that of (27) however good the theory may be, first because
not all of the impacts are central (page 596), then because in most (not
quite all) of the experiments the bombarded substance is in a thick
layer instead of a thin film (page 597), and finally because of the
functions /i and f^ of equation (20). Much as we should like for
simplicity to put these functions equal to unity and zero respectively,
and even though quantum mechanics has removed some of the ob-
stacles to doing so, yet we are obliged to take them into account — the
most striking and cogent reason being that with a variety of elements,
Pt is not the same for impact of protons as for impact of deutons,
though ne is the same for both ! -^

The situation being such, one cannot ask as yet for accurate state-
ments about the values of r^ and F™, the constants of the "crater
model" exhibited in Figs. 12 and 13. These must wait upon a
thoroughgoing fitting of the theory to the experimental curves of
Pt-vs-K, involving a decision as to the magnitude of fi. The values
of Vm for several of the lighter elements have been estimated from the
data on transmutation, but the procedure of arriving at the estimates

12|-
10

6 -

4 -

Mg Al

6 8

ATOMIC NUMBER

Fig. 20— Resonance-levels and (estimated) heights of potential-barriers for some
of the lighter elements, deduced from observations of transmutation. (Pollard;
Phys. Rev.)

has not (so far as I know) been published. I reproduce as Fig. 20 a
graph of Pollard's, the circles along the uppermost line showing the
estimated values of eVm and the crosses along the other two lines

^^ Also it has been said that the shapes of the best experimental curves of Pt-vs-K
imply that as K is increased, /i increases at first and then becomes constant; but
there is a great lack of published theory on these matters.

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII

609

showing the values of e F at which resonance occurs. The linear trends
suggest that these may be properties of the nucleus which are suscep-
tible of simple interpretations. There are also the estimates of Vm
made from observations on alpha-particle scattering, most of which
are merely minimum-admissible-values below which Vm cannot lie,
while a few are more definite. As for Vm, there is at any rate nothing

5 222

O
I-
<

218

85 86 87 88

ATOMIC NUMBER

Fig. 21 — Genealogies of the radioactive elements. (The actinium series is plotted
some distance above the others for legibility, but almost certainly An should lie one
unit below Tn, the rest correspondingly).

to indicate that we must make it higher than the values — a few times
10~^^ cm. — which many reasons impel us to assign to the dimensions
of nuclei.

Thus the quantum-mechanical theory of transmutation is as yet

610 BELL SYSTEM TECHNICAL JOURNAL

in a primitive state, and indeed not advanced enough (in my estima-
tion) to be considered fully proved by its own successes. Quantum
mechanics has, however, many other buttresses, quite sufficiently
many to allow us to take it for granted ; and in this particular field it
has the prestige of prophetic powers. Until the experiments of Cock-
croft and Walton, no one had ever effected transmutation except with
alpha-particles of charge 2e and energy K amounting to several
millions of electron-volts. Now Cockcroft and Walton say that they
were encouraged to build the elaborate apparatus necessary for trying
it with protons of energy much less than one million electron-volts,
by Gamow's inference that particles of charge -\- e should have a
very much greater chance of penetrating through a potential-hill and
into a nucleus, than particles of equal kinetic energy and only twice
the charge — the inference from the fact that ne occurs in the exponent
of the exponential function appearing in equation (23) and others
like it. Moreover, the phenomenon of resonance was predicted by
Gurney (and mentioned by Fowler and Wilson, who, however, appar-
ently did not believe that it could ever be observed) before it was
discovered in the experiments of Pose.

The merits of the crater model with the quantum-mechanical theory
have, however, not yet been fully presented, for I have left to the last
their application to radioactivity.

One of the principal features of radioactivity — both the "induced"
variety described in the early part of this article, and the "standard"
variety known these thirty-five years — is the exponential decline or
decay of the intensity, hence of the quantity of any radioactive
substance, as time goes on. This signifies that the average future
duration, reckoned from any instant of time, of all the atoms surviving
unchanged at that instant, is the same whichever instant be chosen —
or, that the probability that an atom, not yet transformed at instant
/o, shall undergo its transformation within (say) a second of time
beginning at /o, has the same value however long the atom may have
existed up to this arbitrarily-chosen-moment /q.

All this is commonly expressed by saying that radioactive trans-
formations obey the laws of chance. I quote (not for the first time)
a passage from Poincar^, which illustrates how this had to be inter-
preted before the advent of quantum mechanics; I take the liberty
of writing "nucleus" where he wrote "atom":

". . . If we reflect on the form of the exponential law, we see
that it is a statistical law; we recognize the imprint of chance. In this
case of radioactivity, the influence of chance is not due to haphazard
encounters between atoms or other haphazard external agencies. The

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 611

causes of the transmutation, I mean the immediate cause as well as
the underlying one {la cause occasionnelle aussi Men que la cause pro-
fonde) are to be found in the interior of the atom fread, in the nucleus!] ;
for otherwise, external circumstances would affect the coefficient in
the exponent. . . . The chance which governs these transmutations
is therefore internal; that is to say, the nucleus of the radioactive
substance is a world, and a world subject to chance. But, take heed !
to say 'chance' is the same as to say 'large numbers' — a world built
of a small number of parts will obey laws which are more or less com-
plicated, but not statistical. Hence the nucleus must be a com-
plicated world." 2^

Well! the advent of quantum mechanics has made unnecessary
the conclusion which Poincare was obliged to draw; for according to
this doctrine, the statistical law is characteristic as much of a single
particle confronted with a potential-hill, as of the greatest conceivable
number of particles mixed up together. It must be admitted that
Poincare's conclusion is probably right enough for the radioactive
nuclei of which he knew, all of which must be conceived to comprise
several hundreds of particles, protons and electrons and neutrons and
the like; but it is not enforced by the reason which he gives, if quantum
mechanics is valid. For reversing the argument of previous pages:
if in the valley-enclosed-by-hills which is illustrated (for the oversim-
plified one-dimensional case) by Fig. 16, we postulate a particle and
the waves associated with that particle, then the quantum-mechanical
boundary-conditions require waves beyond the hills as well, and the
coexistence of waves without and within implies a tendency — -a
tendency governed by the "laws of chance," a probability — for the
particle to escape from within to without. As soon as the physicist
has successfully made the effort of consenting to quantum mechanics,
he is dispensed from the further effort of contriving nuclear models
with special features to account for the law of decay of radioactive
substances.

Like the probability of entry, the probability of escape of the
particle from the confined valley is governed by the ratio of the
squared amplitudes -^^if* within the valley and beyond the hill (the
latter in the numerator). It thus is governed by the exponential
function,

exp. [- (47r//?.) f^2m{NeVm - E)dx'],

of which we have already made the acquaintance, multiplied by a

2^ H. Poincare: "Dernieres Pensees," pp. 204-205 (he credits Debierne with the
idea).

612 BELL SYSTEM TECHNICAL JOURNAL

factor </)(£) which itself may be a function of E the energy of the
particle, but is of secondary importance. Such a formula, in the
case of penetration from without, represented the probability of entry
for a single approach of the particle to the hill. This suggests that
we should deem it in this case as representing the probability of escape
for a single approach of the particle from the depths of the valley to
the inner side of the hill. Suppose the valley to be of breadth a, the
particle to be bumping back and forth in it with speed vc the number
of approaches of the particle per unit time to the hill will be equal to
Vila. If the bottom of the valley is at the same level as the axis of
abscissae in Fig. 16, Vi is equal to ^2Elm\ if the valley is deeper, Vi is
greater. We deduce for the mean sojourn of the particle within the
valley, which is the reciprocal of the probability-of-escape-per-unit-
time, the expression :

[(t;i/a)0 (£)]-' exp. [+ (47r//0 ( ^2m{NeV,n - E)dxJ (28)

The aspect of this expression is far from encouraging to one who
wishes for a striking quantitative test of the theory. Its value de-
pends not merely on the breadth a assumed for the space within the
potential-hill and the height Vm of the hillcrest, but on the details of
the shape assumed for the potential-curve of Fig. 12 both within and
without the crest; and since there is little or no independent knowledge
of these qualities of the nucleus, they may be adjusted practically at
will to fit any observed value of T whatever. Furthermore it was
obtained by making certain crude assumptions and certain not very
close approximations.

One essential test, however, can be applied to it, which it must pass;
and pass it does. Let values of a and Vm, and a shape for the potential-
hill of Fig. 12, be so chosen that for some particular radioactive ele-
ment, RaA for instance, equation (28) agrees with experiment; which
is to say, that when into the right-hand member of (28) is substituted
for E the observed kinetic energy of the emerging alpha-particles, the
value of this right-hand member becomes equal to the observed mean
life of the element. Now let precisely the same values of a and Vm
and the same shape of hill^" be assumed for some other radioactive
element, RaC for instance; in the right-hand member of (28), let the
observed kinetic energy of the (main group of) alpha-particles for
RaC be substituted for E; and let the value of T be computed. We

^^ Excepting that the two hills should be expected to slope off towards infinity in
the manners of the two functions Zi/r and Zi/r, where Zie and Zie stand for the
nuclear charges of the nuclei left behind after the alpha-particle departs, and are
often (not always) different for two different radioactive substances.

CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 613

should not expect a perfect agreement, since the two nuclei are not
identical; but we should be disconcerted by a sharp disagreement,
since both nuclei belong to elements of which the nuclear charges
differ at most by only a few per cent and the nuclear masses by little
more. A very great disagreement would in fact be gravely injurious
to the theory. Making the test, Gurney and Condon found, however
that there is no grave disagreement: the theory survives the test.

A very similar test was applied with greater minuteness by Gamow.
For each element he assumed a potential-hill having a vertical rise on
the inward side, and on the outward side a curved slope conforming
exactly to the function (Z — 2)/r, where Z stands for the atomic
number of the element before the alpha-particle quits it and conse-
quently (Z — 2)e stands for the charge of the residual nucleus. In
other words, he postulated a classical inverse-square electrostatic field
("Coulomb field") from infinity inward to a distance Tq from the
centre of the nucleus, and at ro a discontinuous potential-fall. This is
a potential-distribution distinguished by a single disposable constant,
to wit, r^; for Vm itself is determined by Tq and Z.

Gamow proceeded to compute what value must be assigned to ^o
in order to achieve agreement between theory and experiment for
each of the twenty-three alpha-emitters. Approximations must be
made in carrying through the calculations; those which Gamow em-
ployed convert equation (28) into this:

loge r = - loge {h/4:mro~) + 87rVVm (Z - 2)//W2£

-f (167rgVm/;0VZ^^Vr^.

Putting for E the kinetic energy of the alpha-particles and for T
the mean lives of the several elements, he evaluated ro. Had the
values proved very different for the various alpha-emitters, it would
have spoken ill for the theory; but all the values were comprised be-
tween 9.5 and 6.3 times 10~^^ cm. The order of magnitude is satis-
factory; the differences between the several values are by no means
disagreeably great; and there are even signs of a systematic upward
trend of the values of ro with the atomic numbers of the nuclei. The
quantum-mechanical theory and the crater model of the nucleus so
pass their crucial test.

The Measurement and Reduction of Microphonic Noise
in Vacuum Tubes

By D. B. PENICK

The microphonic response of different types of vacuum tubes to the same
mechanical agitation covers a 70 db range of levels. Tubes of the same type,
on the average, cover a range of about 30 db. These response levels are
too sensitive to minute variations in testing conditions to be measurable
with any great precision, but values which are reproducible to within 5 db
are obtainable with a laboratory test set comprising a vibrating hammer
agitator, a calibrated amplifier, and a thermocouple galvanometer indicator.
Sputter noise is made measurable by frequency discrimination methods.

Minimum microphonic disturbance under given service conditions is
attained by using the less microphonic types of tubes which are available, by
selecting the quieter tubes of a given type for use in positions sensitive to
mechanical disturbance, and by protecting the tubes from mechanical and
acoustic vibration. Examples of quiet triodes are the Western Electric
No. 264B (filament) and No. 262A (indirectly heated cathode). Indirectly
heated cathode type tubes are intrinsically less microphonic than filamentary
types. Further microphonic improvement in the tubes themselves is made
difficult by requirements for favorable electrical characteristics. Well
designed cushion sockets can reduce microphonic levels by as much as 30 db,
and other methods of cushioning, more expensive and less compact, can
extend the reduction even farther. Sputter noise can be eliminated almost
entirely in most types of tubes by commonly applied design features and
manufacturing methods.

A MAJOR problem which has had to be met by every engineer who
has designed a high gain ampHfier is that of eHmination or reduc-
tion of noise. Noise of one kind or another, extraneous to the desired
signal, is always present in any amplifier, and sets a lower limit on the
smallness of the signal which can be amplified without intolerable
interference. In many experimental and commercial amplifiers, the
technical and economic obstacles to noise reduction necessitate a
compromise between inherent noise level and sufficient volume range
for ideal reproduction. The possible sources of this noise are numerous
and include power supply, faulty contacts, insulation leaks, pick-up
from stray fields, and many other disturbing elements. Among the
most persistent types of noise, however, requiring particularly careful
design for their elimination or satisfactory reduction, are three which
originate in the vacuum tubes themselves, namely, fluctuation noise,
microphonic noise, and sputter noise.

Fluctuation noise has been treated at some length by Schottky,^

1 W. Schottky, Ann. der Phys., v. 57, p. 541, 1918; v. 68, p. 157, 1922. Phys. Rev.,
V. 28, p. 74, 1926.

614

MICROPHONIC NOISE IN VACUUM TUBES 615

the discoverer of one of its sources and by several other investigators.^
It is the fundamental noise arising from the circumstance that the elec-
tron current is a stream of discrete particles rather than a continuous
flow. For ordinary types of low power tubes of good design, over the
audio band of frequencies, the root-mean-square amplitude of this
noise is equivalent to about 1 microvolt (120 db below" 1 volt) of noise
voltage applied to the grid of the tube. G. L. Pearson ,2 in a recent
paper, has pointed out that for the best signal-to-noise ratio, the input
impedance should be so large that the thermal noise arising in this
impedance predominates over the fluctuation noise arising in the tube.
In many broad-band or high-frequency systems, however, such an
ideal condition is practically unattainable, and fluctuation noise
remains as a limiting factor.

It is with the second type, microphonic noise, that we are particu-
larly concerned here, though sputter noise is also of interest and will
be dealt with briefly in a later paragraph. Microphonic noise, as the
name is usually applied, is the familiar gong-like sound which is always
produced when a vacuum tube, followed by a sufficiently high-gain
amplifier and a sound reproducer, is subjected to a mechanical shock.
Its origin is in the vibrations of the various elements of the tube, which
make minute, more or less periodic changes in the spacings of the ele-
ments and therefore make corresponding changes in the plate current,
whose value at every instant depends on these spacings. Its intensity
in a given tube depends on the type and intensity of agitation to which
the tube is subjected. For a given agitation, microphonic noise may
be reduced either by stiffening and damping the tube structure, thereby
reducing the amplitude and duration of vibration of the elements, or
by cushioning the tube so that it receives only part of the original
agitation.

In order to treat the problem of noise reduction intelligently, it is
necessary to have a measure of the effectiveness of treatments applied.
To this end, the properties of microphonic response in vacuum tubes
have been studied, and a test set has been designed and built for labora-
tory use which affords a quantitative measure of microphonic response
in tubes, and of effectiveness of cushioning in cushion sockets. Some
of the more important characteristics of microphonic noise will now be
considered.

^"The Schottky Effect in Low Frequency Circuits," J. B. Johnson, Phys. Rev.,
V. 26, pp. 71-85, July, 1925.

"A Study of Noise in Vacuum Tul)es and Attached Circuits," F. B. Llewellyn,
Proc. I.R.E., V. 18, pp. 243-265, Feb., 1930.

"Shot Effect in Space Charge Limited Currents," E. W. Thatcher and N. H.
Williams, Phys. Rev., v. 39, pp. 474^96, Feb. 1, 1932.

"Fluctuation Noise in Vacuum Tubes," G. L. Pearson, Physics, v. 5, p. 233,
September, 1934. Also published in this issue of Bell Sys. Tech. Jour.

616 BELL SYSTEM TECHNICAL JOURNAL

Factors Affecting Microphonic Noise Levels
In the first place, it may be pointed out that the production of micro-
phonic noise in commercial types of vacuum tubes is an extremely
complicated phenomenon. Each individual component of the mechan-
ical structure is a complete vibrating system having several modes of
vibration and natural resonant frequencies, and usually very little
damping as compared with electrical circuits. These components, all
coupled together mechanically in various ways, form a mechanical
network much more complex than the electrical networks encountered
in communication engineering practice.

The complexity of the mechanical vibration is reflected in the com-
plex character of the noise itself, and is admirably illustrated by the
frequency-response characteristics published by Rockwood and
Ferris,^ and by similar characteristics obtained in the course of this
work. It is further demonstrated by the experimental fact that when
a large group of supposedly identical tubes is tested by applying the
same mechanical vibration to each tube in turn, mounted in the same
socket, the response levels of individual tubes may differ from each
other by as much as 30 db for representative types of tubes. Such a
magnitude of variation would not be expected to result from the
comparatively small dimensional variations tolerated in manufacture,
and must be explained by the exaggerating effect of intercoupled me-
chanical resonances in a complicated vibrating system. A curve
showing a typical distribution of microphonic response levels in a
group of tubes of the same type measured under identical conditions of
agitation is shown in Fig. 1. The general shape of this curve is char-
acteristic of any function subject to random variations about a mean,
and the range of levels included between the quietest and the noisiest
tubes, approximately 30 db, is about average for different types of
tubes. Tubes of exceptionally firm construction and of fairly wide
spacing, may vary over as small a range as 15 or 20 db, while tubes in
which there is a possibility of slight looseness of parts, may vary over
as great a range as 40 db or even more.

The nature and intensity of the agitation, the vibrational charac-
teristics of the tube mounting, and the type and degree of mechanical
coupling between the tube and its mounting also play an important
part in the determination of the microphonic response of a particular
tube, since the tube and its closely coupled mounting make up a single
vibrating system. The reason for considering the coupling apart
from the mounting is that it is usually of the pressure-friction variety
and is subject to random variation.

' "Microphonic Improvement in Vacuum Tubes," Alan C. Rockwood and Warren
R. Ferris, Proc. I.R.E., v. 17, pp. 1621-1632, Sept., 1929.

MICROPHONIC NOISE IN VACUUM TUBES

617

< 70

O 50

0 5 10 15 20 25 30 35 40

MICROPHONIC LEVEL IN DECIBELS BELOW 1 VOLT

Fig. 1. — Typical distribution of microphonic noise levels produced by a constant,
artificial, mechanical stimulus (Western Electric No. 102F Vacuum Tube).

It is found experimentally that with no type of commercial socket
which has been tested can a tube be removed and reinserted, or even
be left in the socket for a period of time subject to incidental jars and
temperature fluctuations, with the expectation of perfectly reproducing
a previously measured microphonic level. The sort of random varia-
tion which is usually found is illustrated in the reproducibility chart of
Fig. 2. In this chart, each point represents two separate observations
of microphonic level made on the same tube in the same apparatus, the
tube having been removed and reinserted between the two observa-
tions. The two levels thus measured are represented by the abscissa
and ordinate, respectively, so that if the measurements were perfectly
reproducible, all of the points would lie on a straight line making an
angle of 45 degrees with the coordinate axes and passing through the
origin. The amount of maximum scattering here is about 5 db and
may be considered an average value. In some cases, with commercial

618

BELL SYSTEM TECHNICAL JOURNAL

<n 16

'C 14

o
o

/
/

c
o

°s

o°oO

/■

/£ o

>-<

5
3 ° c

o
o

X O

° °o

10 12 14 16 18 20 22 24 26 28 30 32 34 36

SECOND test: microphonic level in decibels below 1 VOLT

Fig. 2 — Reproducibility of microphonic noise level measurements using a commercial
socket with a constant, artificial, mechanical stimulus (100 No. 102F Tubes).

sockets, it has been observed to be as low as 3 db and in others as high
as 8 db.

In order to show that this random variation is not due to the tube
itself, experiments have been made with two forms of suspension which
minimize the reaction of the mounting on the vibration of the tube and
so reduce as far as possible the effect of variation in coupling. In one
set-up, the tube is hung by a single thread of rubber, stretched to its
elastic limit, the electrical connections being made by very light,
flexible leads fastened with light clips directly to the prongs of the
base. In the other, the tube is clamped lightly between two large
blocks of very soft sponge rubber, and the electrical connections are
made through rhercury cups into which the base prongs dip. In both
cases, the agitator is a light pendulum striking the base or bulb of the
tube. The two mountings give very similar results, and are charac-

MICROPHONIC NOISE IN VACUUM TUBES

619

terized by very much less scattering than any normal tube mounting,
as may be seen in the correlation chart of Fig. 3, which is typical of all
of the tests made with these light suspensions. The maximum scatter-
ing here is only about 1 db.

Going to the opposite extreme in tube mounting, similar tests have
been made with the tube base held tightly in a split metal clamp,

44
42
40

)

/»

-/"

y°

/°

/s>

26 26 30

SECOND TEST :

32 34 36 38 40 42 44

MICROPHONIC LEVEL IN DECIBELS BELOW I VOLT

Fig. 3 — Reproducibility of microphonic measurements using a rubber clamp tube
mounting with a constant, artificial, mechanical stimulus (37 No. 102F Tubes).

which itself is bolted rigidly to a heavy base. As is to be expected, the
observed levels vary widely and erratically for successive insertions of
the tube, and the mere tightening or loosening of the thumb-screw
controlling the pressure of the clamp on the base in some cases changes
the level by as much as 10 db.

As for the nature of the applied agitation and the vibrational char-
acteristics of the tube mounting, a countless number of combinations

620

BELL SYSTEM TECHNICAL JOURNAL

of these exists, each of which would agitate the tube in a different way.
However, from tests made with a variety of mounting arrangements
for the tube under test and a variety of degrees of intensity and points
of appHcation of forms of impact agitation, it may be concluded that in
practical set-ups these factors may be varied widely without changing
the general nature of the microphonic level measurements greatly.
That is, the form and breadth of the distribution curve and the scatter-
ing of the points on the reproducibility chart for any typical group of
tubes are likely to be quite similar to Figs. 1 and 2, respectively, for
almost any practical impact agitator.

Although the general nature of the results obtained with various
combinations of these agitator and mounting arrangements is about the
same for all of them, there are certain particular dififerences, which
show up chiefly in two characteristics. One is that the mean noise
level of a group of tubes is in general not the same for different mount-
ings and methods of agitation. That this must be true is fairly ob-
vious and needs no comment. The other is illustrated in Fig. 4,
which is a correlation chart showing typical results of measurements

CO
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0
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Fig.

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

TEST ON APPARATUS RACK

MICROPHONIC LEVEL IN DECIBELS BELOW 1 VOLT

4 — Comparison of two tube mountings with a constant, artificial, mechanical
stimulus (100 No. 102F Tubes).

MICROPHONIC NOISE IN VACUUM TUBES

621

of the same group of tubes on two different agitating systems. One
system in this case consists of a rectangular slate block vibrated by
repeated blows of an electrically operated hammer. The other system
consists of a steel panel carrying the tube under test, mounted on an
apparatus rack which is vibrated by a single blow from a steel ball
falling as a pendulum against the rack. The points on this chart
scatter about an ideal line over a band about twice as broad as that
in Fig. 2 where a test is made and repeated on the same testing unit.
It may also be observed that the mean noise levels produced by the
two systems are different, about 35 and 43 db below one volt respec-
tively.

The effect of varying the intensity of agitation is shown in Fig. 5.

O30

I 35

Q
Z 50

-o

^^^

-o-

-f

•

—

•

y
y'

O 65

2 70

50 100 500 1000 5000 10,000

IMPACT MOMENTUM IN GRAM CENTIMETERS PER SECOND

Fig. 5 — Effect of intensity of agitation on 4 No. 264B Tubes.

The four curves represent four No. 264B Tubes tested under the same
conditions. In making the measurements, the tube under test is
mounted in an ordinary socket on a heavy base, which is agitated by
means of a pendulum swinging against it and making one rebound.
From measurements of the initial swing of the pendulum, its rebound,
and its mass, the total momentum imparted to the tube mounting
during the impact can be calculated. This quantity is plotted as
abscissa in the figure and is proportional to the initial velocity imparted
to the tube mounting at the point of impact. Different values of
momentum are obtained by varying the initial swing and the mass of
the pendulum. At the lower values of momentum, the observed

622

BELL SYSTEM TECHNICAL JOURNAL

points lie, within the limits of experimental error, on parallel straight
lines -so drawn that along them the microphonic noise level expressed
in volts is proportional to the initial velocity of the tube mounting.
Some such relation as this would be expected to hold as long as the
response of the system is linear. The departure from this law at higher
values of momentum, then, probably indicates non-elastic motion
either of elements of the tube with respect to one another or of the tube
with respect to the socket. It may be noted in passing that the No.
264B Tube is exceptionally rigid in structure and that in more loosely
constructed tubes, the straight line part of the response curve ends
at much lower intensities of agitation.

Since the noise energy is spread over a band of frequencies, the
microphonic response observed in any given reproducing system de-
pends also on its frequency-response characteristic. In the usual type
of volume indicator, the response is substantially uniform over the
audio range of frequencies, but where the final auditory sensation is
being considered, the overall characteristic is modified by that of the
ear of the listener.^ The effect of changing the overall response
characteristic is illustrated for one particular case in Fig. 6 and Table I.

0-I5

^-20
O

01-25

^-^

\ ~~~

(a?"

(b)

500 1000 5000

FREQUENCY IN CYCLES PER SECOND

Fig. 6 — Microphonic noise amplifier frequency characteristics.

10,000

Here, two sets of measurements have been made on each of three types
of tubes, one set using an amplifier having a fairly uniform gain char-
acteristic, curve (a), Fig. 6, and the other set using a weighted ampli-
fier, weighted as in curve (b) in the figure. The same agitator and

^ "Speech and Hearing," H. Fletcher. D. Van Nostrand Co., 1929.

MICROPHONIC NOISE IN VACUUM TUBES

623

TABLE I
Microphonic Levels in db Below 1 Volt

Type Tube

Amplifier (a)

Amplifier (6)

Difference

264B

102F

231D

37.9
29.7
18.2

63.2
44.7
38.9

25.3
15.0
20.7

indicator are used in both sets of measurements. Table I gives the
mean noise levels for each of the three types of tubes and the differences
between the values obtained for each type with the two amplifiers.
The results represent about ten tubes of each type. The weighted
amplifier, of course, gives the lower levels for all tubes since the noise
components at all frequencies except 1000 c.p.s. are amplified less by
this amplifier than by the more uniform amplifier. The magnitude
of the difference in level depends on the frequency spectrum of the
microphonic noise being measured and in general is different for differ-
ent types of tubes as in this illustration.

Still another important factor which affects the microphonic response
of a given system is the relation between the rate of variation of the
noise intensity and the time-response characteristic of the system as a
whole, usually determined by the indicator. The indicator may be a
meter, oscillograph, or other device, or it may be the ear of a listener.
A slow moving indicator would respond less to a pulse of noise, such as
might be produced by a single shock to a tube, than a more rapidly
responding indicator having the same sensitivity to a steady signal.
The time required for the ear to reach its maximum response to a
suddenly applied sound is about 0.2 second.^

The degree of importance of the time-response characteristic of the
indicator in measuring transient pulses may be inferred from Table II.
This table gives the results of two sets of measurements made on the
same three groups of tubes with a single impact type of agitator. The

TABLE II
Microphonic Levels in db Below 1 Volt

Type Tube

0.2 Second Indicator

2.0 Second Indicator

Difference

264B

102F

231D

37.9
29.7
18.2

50.5
37.2

25.2

12.6

7.5
7.1)

^"Theory of Hearing; Vibration of Basilar Membrane; Fatigue Effect," G. V.
Bekesy, Physikalische Zeilschrift, v. 30, p. 115, March, 1929.

624 BELL SYSTEM TECHNICAL JOURNAL

two sets differ only in the indicators used. One requires approximately
2 seconds to reach its maximum deflection with a steady impressed
signal, and the other requires about 0.2 second. The differences in
level corresponding to different types of tubes do not vary greatly,
but are nevertheless appreciable. They are, to some extent, a measure
of merit of the tube, for a larger difference indicates higher damping of
the microphonic disturbance, and high damping is of course desirable.

A Microphonic Noise Measuring Set

The type of test set which has been built in the course of this study
for use in the laboratory, comprises an arbitrary standard of agitation,
a calibrated amplifier, and an indicating instrument. The agitator
consists of a heavy, rectangular slate base at one end of which are
mounted sockets for several types of tubes. At the other end is an
electrically driven vibrating armature carrying a hammer which
strikes about 9 blows per second against a steel block bolted firmly near
the center of the slate base. This unit is set on a thick felt pad in a
felt lined copper box which provides electrical shielding and some de-
gree of sound-proofing. The sockets used (except those for the bay-
onet-pin bases) are of the type in which contact springs push each base
prong firmly to one side, against the body of the socket. This type
has been found to stand up well under repeated insertions and with-
drawals of tubes and gives as good correlations between repeated micro-
phonic measurements as any type which has been tried.

The amplifier is basically a simple resistance-choke coupled unit
having a frequency-response characteristic which is essentially flat
(within 3 db) between 80 and 30,000 c.p.s. The tube under test,
whose plate voltage is supplied through an 80-henry choke, works
directly into a 100,000-ohm potentiometer, variable in 10 db steps,
whose output is connected to the input of the amplifier. The indicator
is a sensitive thermocouple galvanometer whose scale is marked off
in db and half db divisions so that the noise level may be read directly
from the setting of the input potentiometer and the position of the
indicator. It has been found convenient to think of the noise level in
terms of the root-mean-square voltage developed by the tube across
the 100,000'Ohm load resistance and to use 1 volt as the reference level.
Accordingly, unless otherwise noted, the noise levels given herein are
expressed as db below 1 volt across a 100,000-ohm load resistance.

In order to correct for time shifts in tube characteristics and battery
voltages, provision is made for checking the amplifier calibration at
any time by throwing a switch which transfers its input circuit from
the tube under test to a local oscillator. This oscillator delivers a

MICROPHONIC NOISE IN VACUUM TUBES 625

small, fixed output voltage which is measured and set at a predeter-
mined value with the aid of another thermocouple galvanometer.
The amplifier gain may be adjusted, by means of a small range, con-
tinuously variable potentiometer until the indicator gives the proper
reading to correspond with the known level of the applied input.
With reasonably steady battery voltages, this calibration is necessary
only two or three times in the course of a day's testing. The range of
noise levels for which the amplifier is calibrated extends from 10 db
above 1 volt to 65 db below 1 volt. This range has been found to
include practically all tubes which it has been desired to test with the
standard agitator.

The flat amplifier characteristic, which has been described, is nor-
mally used for general testing in connection with vacuum tube design
work since it gives the highest microphonic level readings and there-
fore the most conservative picture of the performance of the tube from
the standpoint of the designer. Provision is made, however, for
switching in a specially designed weighted amplifier such as is used in
making routine noise measurements in telephone speech circuits.®
The frequency characteristic including this unit has already been
shown in Fig. 6, curve (h), and is designed to compensate for the in-
terfering effect of each component of noise on the average ear plus the
effect of the frequency characteristic of the telephone subset. A similar
weighting network compensating for the non-uniform frequency re-
sponse of the ear alone would also be useful, but has not yet been
provided.

Nature and Measurement of Sputter Noise

By making a slight modification of the amplifier circuit, this test
set may also be used to measure sputter noise. Sputter noise is a
descriptive name applied to a class of noises characterized by a harsh
crackling or sputtering sound easily distinguished from the gong-like
quality of microphonic noise or the steady roar of electron noise. It
may occur either with or without agitation and is the result of dis-
continuous changes in electrode potential such as may be produced by
imperfect contact between conducting members in a tube or by inter-
mittent electrical leaks across insulation.

Sputter noise due to agitation is always accompanied by micro-
phonic noise, and though it often contains instantaneous peaks of high
intensity which constitute a very disagreeable and annoying type of
interference, its total energy content is usually so small that it contri-

^ "Methods for Measuring Interfering Noises," Lloyd Espcnschied, Proc. I.R.E.,
V. 19, pp. 1951-54, Nov., 1931.

626

BELL SYSTEM TECHNICAL JOURNAL

butes very little to the ordinary microphonic level reading. Special
methods must therefore be used in order to make measurements of
sputter noise which are independent of microphonic noise. One
method which has been found to be effective and convenient, is that of
frequency discrimination. If the audio frequency components of the
total noise are cut out, then microphonic noise is completely eliminated.
Sputter noise, however, due to its discontinuous character has, theo-
retically, an infinite frequency spectrum, and, practically, one which
extends at least into the broadcast band of radio frequencies.

In the microphonic noise test set, sputter noise measurement is
provided for by switching in a high-pass filter cutting off sharply at
16,000 cycles. Greater sensitivity is also provided by additional
stages of amplification to permit the measurement of the lower levels
found to be characteristic of sputter noise in this frequency range. A
schematic diagram of the microphonic and sputter noise test set is
shown in Fig. 7. The weighted amplifier and the calibrating oscilla-

—

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etui

D in

o Jo

O O UJ

l<^

- a

Jr Q

^°

-I

MICROPHONIC NOISE

SPUTTER NOISE AMPLIFIER-

Fig. 7— Microphonic and sputter noise amplifier schematic diagram.

tors (one for microphonic noise and one for sputter noise) have been
omitted for the sake of clearness.

Reduction of Microphonic Noise

The reduction of microphonic noise from the view-point of the
vacuum tube designer is chiefly a matter of mechanical design and
manufacturing technique. The quietest construction is obviously one
which has the stiffest electrodes and supporting members, the shortest
distances between points of support, and the highest damping of
mechanical vibration. The extent to which these features can be
incorporated in a practical tube design, however, is limited by the
requirements for favorable electrical characteristics. A low filament
current, for example, requires that the filament be small in diameter,
which renders it more susceptible to vibration than a heavier filament ;
or if a tube has an indirectly heated cathode, it is a problem to support

MICROPHONIC NOISE IN VACUUM TUBES

627

the cathode rigidly without conducting away large amounts of heat
along the supports. The diameter, length, and spacing of the grid
lateral wires are fixed within relatively narrow limits when the desired
values of amplification factor and internal impedance are fixed, pre-
cluding any important increase in stiffness here ; and where high mutual
conductance is desired, it is necessary to use relatively close spacings
between the elements, under which condition a given amplitude of
vibration produces a relatively large per cent change in spacing, and
therefore a high microphonic noise level.

The Western Electric No. 264B Vacuum Tube is an example of
what has been done in working for a stiff, compact structure. The
plate support wires, which also support the whole top of the structure,
are short, straight, and as heavy as is practicable, and an extra wire
from the press braces the glass bead. One of the most important
features of this tube, however, is its filament. In most filament type
tubes the vibration of the filament is the chief source of microphonic
noise. In the No. 264B Tube, therefore, the filament is made com-
paratively short and heavy and is mounted in the form of a broad,
inverted V to whose apex considerable tension is applied by means of a
cantilever spring. The effectiveness of this treatment may be seen
from Table III which lists the mean noise levels for a number of types
of Western Electric small tubes, and the maximum and minimum

TABLE III*

Microphonic Noise Level

in db below 1 volt

No. of
Samples

Class

Type

Tested

Number

Measured output Level

Equivalent

Mean Input

Level

Max.

Min.

Mean

Filament type triode

250

lOlD

833

lOlF

505

102F

235

215A

1,144

231D

201

239A

715

264B

Indirectly heated cathode-

244A

type triode

448

247A

452

262A

Screen grid and pentode

245A

259A

283A

285A

* The microphonic properties of the No. 259A Tube given in this table are identical
with those of the 259B discussed by Pearson.^

628 BELL SYSTEM TECHNICAL JOURNAL

levels which have been observed for each type. The No. 264B, with a
mean level of 42 db below 1 volt, is 20 db quieter than the No. 239A
which it was designed to replace, and is the quietest of the filament
type triodes. The next quietest tube of this structure is the No. 101 D,
in which the elements are supported from a rigid glass arbor and the
filament is quite heavy, requiring one ampere of heating current. The
No. 215A is almost identical with the No. 239A except for a firmer
supporting structure which results in a 5 db improvement. The most
microphonic of the types listed is the No. 231D, which has a very fine
wire filament whose diameter is fixed by the requirement that the heat-
ing current be 0.060 ampere.

If the filament is the chief source of microphonic noise in filament
type tubes, then it is to be expected that tubes having indirectly heated
cathodes will be much less microphonic, inasmuch as the cathode is an
extremely rigid member. An examination of Table III shows that this
is indeed true. The No. 244A and No. 247A types, in which no
special precautions have been taken to obtain quietness, are about as
quiet as the No. 264B Tube. In the No. 262A Tube, therefore, it has
been possible to reduce the microphonic noise still further, to 49 db
below 1 volt, by cementing the elements into rigid supporting blocks
of ceramic material. This tube is also quiet in other respects, notably
in its freedom from AC hum picked up from the cathode heater circuit.''

In comparing tubes having widely different electrical characteristics,
it is not quite fair to compare their noise output levels alone, for given
two tubes having the same noise output, the tube having the higher
gain can be used with smaller signal inputs and have no greater noise
interference in the output. Accordingly, another column is given in
Table III listing the equivalent noise input level which would produce
the observed noise output if the tube itself were perfectly quiet. The
ratio of this value to the signal input level is directly related to the
degree of microphonic noise interference which is effective in the out-
put of the tube. It is computed by adding the voltage gain expressed
in db, of the tube in the measuring circuit, to the microphonic output
level obtained experimentally. The value of this criterion is illus-
trated in comparing the noise interference produced by multi-element
tubes and triodes. Multi-element tubes as a rule have higher noise
output levels than triodes as may be seen by comparing the Nos. 245A,
259A, 283A, and 285A screen-grid and pentode types with the Nos.
244A, 247A, and 262A triodes. When account is taken of the higher
voltage amplification of these former types, however, the noise inter-

' "Analysis and Reduction of Output Disturbances Resulting from the Alternating
Current Operation of the Heaters of Indirectly Heated Cathode Triodes," J. O.
McNally, Proc. I.R.E., v. 20, pp. 1263-83, August, 1932.

MICROPHONIC NOISE IN VACUUM TUBES

629

ference as indicated in the equivalent input noise column of the table,
compares quite favorably with that of the triodes.

From the point of view of the user of vacuum tubes, constrained to
work with available types, the most effective means of microphonic
noise reduction is the use of one of the quieter types of tubes which
have been described. In cases where noise difficulties are experienced
in existing apparatus not readily convertible to the use of a quieter
type of tube, however, some relief may be gained by selecting the
quieter tubes from a number of the type to be used. To be fully effec-
tive, the selection should be based on measurements made while the
tube is in the socket in which it actually works. Under such circum-
stances, the measurements are reliable to within about 5 db.

Where selection in the field is not feasible, a smaller degree of relief
may still be gained by selection at the factory. The degree of effective-
ness of this method can be deduced from Fig. 4. Suppose, for example,
that quiet tubes for service on the apparatus rack are to be selected by
a test made on the continuous tapper. Choosing the best 25 of the
group as tested by the continuous tapper (those plotted above the
horizontal line in the figure), it is immediately obvious that when these
selected tubes are tested on the apparatus rack (compare abscissae in
Fig. 4) the worst tubes are somewhat quieter than some of those in the
remaining portion of the group. This is more clearly shown in Fig. 8

J 90

$ 80

z
<

£ 70

< 60

O 30

WHOLE GROUP y

/•

3EST 25 TUBES
AS SELECTED

3Y CONTINUOUS
TAPPER TEST

34 36 38 40 42 44 46 48 50 52

MICROPHONIC LEVEL MEASURED ON APPARATUS RACK
(PENDULUM AGITATOR) IN DECIBELS BELOW I VOLT

Fig. 8 — Effectiveness of selection of quiet tubes.

630 BELL SYSTEM TECHNICAL JOURNAL

in which the distribution of levels in this group of 25 tubes is compared
with that of the total group as tested on the apparatus rack. The
noisiest tubes in the selected group are from 6 to 8 db quieter than the
noisiest tubes of the unselected group, and in the selected group,
there are none of the tubes which make up the worst 18 per cent of the
whole group.

In several commercial situations where microphonic disturbance
was at one time troublesome, this type of selection has proved to be of
practical value. In these situations, the number of quiet tubes re-
quired is only a small percentage of the manufactured output. Fur-
thermore, only a small percentage of the normal output of tubes are
found to be prohibitively noisy. Under such circumstances, it is
found that when selected tubes from the quietest 25 per cent of the
manufacturers stock are used in the positions most sensitive to mechan-
ical shock, the disturbance in these types of equipment either disap-
pears entirely or recedes to such a level that it is no longer troublesome.

Protection from Shock

Where selection of quieter tubes is not feasible or is not sufficiently
effective, further reduction of microphonic noise may be achieved by
protecting the tube from mechanical and acoustic shock. A very
efficient agency for protection from mechanical shock is a well-designed
cushion socket. The effectiveness of such a socket depends on its
vibrational transmission characteristics considered in relation to the
response characteristics of the tubes used. Considerable improvement
is usually obtained, however, whatever the combination of tube type
and socket type. Figure 9 shows two typical cases of microphonic
improvement obtained by using one of several good types of cushion
socket which have been tested. The curves drawn in solid lines repre-
sent the distributions of microphonic noise levels of a group of No.
102F Tubes tested in one instance in a rigid socket, and in the other
in a cushion socket. The mean improvement here due to the cushion
socket, is about 30 db. The dotted curves represent similar tests
made on a group of No. 262A Tubes and show a mean improvement of
about 18 db.

In cases where the noise must be reduced to very low levels, it may
not be sufficient to protect the tube from disturbances transmitted
mechanically through its base and socket. Except in a perfectly quiet
location, there is always some disturbance produced by sound waves
impinging directly on the bulb of the tube. Ordinarily this disturbance
is negligible, but where the base is sufficiently well cushioned, it may
be of controlling importance. It can be reduced only by reducing

MICROPHONIC NOISE IN VACUUM TUBES

631

^ 90

o
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5 80

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1
1

TYPE I02F TUBE

TYPE262A TUBE

/
/

1
1/

/
/

Sj 60

O 50

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/

1
I

/
/

(/>
O

RIGID
SOCKET

1
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CUSHION^
SOCKET

■/,'

1-
5

1
1

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£ 10

t
/
/

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40 45 50 55 60 65 70 75 80

MICROPHONIC LEVEL IN DECIBELS BELOW I VOLT

Fig. 9 — Effect of cushion socket.

the intensity of the sound wave which is finally allowed to reach the
tube, by some such means as enclosing the tube in a heavy, air-tight
container.

Reduction of Sputter Noise

The reduction of sputter noise in vacuum tubes is chiefly a problem
for the tube manufacturer. Where sputter noise exists in a tube, and
exists only with agitation, it is often eliminated by the same cushioning
measures which are applied to reduce microphonic noise, but in many
cases, satisfactory reduction of sputter would require prohibitive
amounts of cushioning. Fortunately, however, the known design
features and manufacturing methods, which are now generally applied
to tubes of good design, are for the most part quite effective in reducing
sputter noise to a negligible level. In the older types of filamentary
tubes, for example, sputter noise was often present due to the rattling
of the fi,lament at the hook supports at operating temperatures. This
source of sputter has been removed in most present day tubes by keep-
ing the filament under tension at all times by means of flexible canti-
lever spring supports. The effectiveness of this treatment is illustrated
in Fig. 10, which shows distributions of sputter noise levels for two

632

BELL SYSTEM TECHNICAL JOURNAL

Zi 60

O 50

> 40

STIFF >
FILAMENT /
HOOKS /

SPRING
FILAMENT ,
HOOKS /

50 55 60 65 70 75 80 85 90

SPUTTER NOISE LEVEL IN DECIBELS BELOW I VOLT

Fig. 10 — Effect of filament loosenees on sputter noise.

groups of tubes identical in every respect except that one group has
spring filament supports while the other has the older rigid supports.
In 80 per cent of the tubes, the improvement in the sputter noise is
from 20 to 25 db when the spring hook is used.

The source of sputter noise most difficult to control in present day
tubes is insulation leaks. These are commonly due to very thin films
of conducting material which have been deposited on the surface of the
insulating members by sputtering or evaporation during the exhaust
or operation of the tube. Experience has shown the conductivity of
these films to be intrinsically unstable and discontinuously variable.
This condition alone can and does produce sputter noise, but to make
matters worse, the metal support wires of the tube are often in only
loose contact with the insulating parts and the conducting films cover-
ing them so that mechanical agitation breaks and makes the contact
and increases the intensity of the noise. The reduction of these insula-
tion leaks is largely a matter of choice of materials, of manufacturing
technique to reduce the evaporation of conducting material during
exhaust, and of mechanical design to shield important surfaces from
contamination during the normal operation of the tube. Great prog-

MICROPHONIC NOISE IN VACUUM TUBES 633

ress has been made in recent years in effecting an adequate reduction
of leaks economically, and in applications where requirements for
exceptionally low noise levels warrant slightly increased manufacturing
costs, almost any degree of reduction of leaks may be obtained.

Conclusion

The methods which have been outlined for reducing microphonic
noise by cushioning and by making use of the quiet tubes which are
available are, for the present, adequate to meet all but the most ex-
treme requirements. Should the necessity for further reduction be-
come sufficiently urgent in the future, however, it can probably be ob-
tained either by designing still quieter tubes or by improving the cush-
ioning of sockets. The latter course appears to be the more economical.
In either case, however, greatest effectiveness can be attained by con-
sidering particular types of tubes and sockets in their relation to one
another.

The author is greatly indebted to Drs. M. J. Kelly and H. A. Pidgeon
for their kind cooperation and many helpful suggestions in the course of
this work.

Fluctuation Noise in Vacuum Tubes *

By G. L. PEARSON

The fluctuation noises originating in vacuum tubes are treated theoret-
ically under the following headings: (1) thermal agitation in the internal
plate resistance of the tube, (2) shot effect and flicker effect from space current
in the presence of space charge, (3) shot effect from electrons produced by
collision ionization and secondary emission, and (4) space charge fluctuations
due to positive ions. It is shown that thermal agitation in the plate
circuit is the most important factor and should fix the noise level in low
noise vacuum tubes; shot noise and flicker noise are very small in tubes
where complete temperature saturation is approached; shot noise from
secondary electrons is negligible under ordinary conditions; and noise from
space charge fluctuation due to positive ions is usually responsible for the
difference between thermal noise in the plate circuit and total tube noise.

A method is deduced for the accurate rating of the noise level of tubes in
terms of the input resistance which produces the equivalent thermal noise.
Quantitative noise measurements by this method are reported on four difi'erent
types of vacuum tubes which are suitable for use in the initial stage of high
gain amplifiers. Under proper operating conditions the noise of these
tubes approaches that of thermal agitation in their plate circuits at the
higher frequencies and is 0.54 to 2.18 X 10"'^ mean square volts per cycle
band width in the frequency range from 200 to 15,000 cycles per second.
Below 200 cycles per second the noise is somewhat larger.

The minimum noise in different types of vacuum tube circuits is discussed.
These include input circuits for high gain amplifiers, ionization chamber
and linear amplifier for detecting corpuscular or electromagnetic radiation,
and photoelectric cell and linear amplifier for measuring light signals.

With the aid of these results it is possible to design circuits having the
maximum signal-to-noise ratio obtainable with the best vacuum tubes now
available.

Introduction

TT is well known that the noise inherent in the first stage of a high
-^ gain amplifier is a barrier to the amplification of indefinitely small
signals. Even when fluctuations in battery voltages, induction,
microphonic efifects, poor insulation, and other obvious causes are
entirely eliminated, there are two sources of noise which remain,
namely, thermal agitation of electricity in the circuits and voltage
fluctuations arising from conditions within the vacuum tubes of the
amplifier. The effect of thermal agitation in circuits outside the
vacuum tube is well understood, but in the case of tube noise there is
considerable confusion. In order to clarify the whole subject, the
present paper analyzes the various sources of noise in vacuum tubes
and their attached circuits, points out a new method for the measure-
ment of tube noise, reports the results of such measurements on four
different types of vacuum tubes, and discusses the minimum noise in
different types of vacuum tube circuits.

* Published in Physics, September, 1934.

634

FLUCTUATION NOISE IN VACUUM TUBES 635

Often, in the use of high-gain amplifiers, the impedance of the input
circuit is naturally high or may effectively be made high by the use of
a transformer. In this case the contribution of noise from the vacuum
tube is small compared with the noise arising from thermal agitation in
the input circuit. This is a desirable condition since it furnishes the
largest ratio of signal to noise for a given input power. Sometimes,
however, the input impedance is perforce so small that the tube noise
may be comparable with or greater than the thermal agitation noise.
Such conditions may arise, for example, in amplifiers where the frequency
dealt with is high or the frequency range is wide. It is, therefore,
desirable to know the noise level to be expected from different types
of tubes that may be used in the first stage of high-gain amplifiers as
well as to be able to calculate the thermal noise level of the input
circuit.

The noise of thermal agitation ^ arises from the fact that the electric
charge in a metallic conductor shares the thermal agitation of the
molecules of the substance so that minute variations of potential
difference are produced between the terminals of the conductor.
The mean square potential fluctuation is proportional to the absolute
temperature and to the resistive component of the impedance of the
conductor, but is independent of the material. The thermal noise
power is distributed equally over all frequencies although the apparent
magnitude depends on the electrical characteristics of the measuring
system as well as on those of the conductor itself. From purely
theoretical considerations the following equation has been derived ^
to give the thermal noise voltage at the output of an amplifier due to
the thermal agitation of electric charge in an impedance at the input:

R{I)\G,{j)\'df. (1)

Et^ is here the mean square thermal noise voltage across the measuring
device, k is Boltzmann's constant (1.37 X 10~^^ watt second per
degree), T the temperature of the impedance expressed in degrees
Kelvin, R{j) the resistive component of the impedance at the fre-
quency/, Gy{f) the voltage amplification between the input impedance
and the measuring device at the frequency /, and F the frequency
band within which the amplification is appreciable.

While the thermal noise in the circuit is accurately predictable,
the noise originating within the vacuum tube is not completely under-

1 J. B. Johnson, Phys. Rev., 32, 97 (1928).

2 H. Nyquist, Phys. Rev., 32, 110 (1928).

636 BELL SYSTEM TECHNICAL JOURNAL

stood and cannot be calculated accurately. It is known, however,
that tube noise arises from a number of different causes, chief among
which are: (1) thermal agitation in the internal plate resistance of
the tube, (2) shot effect and flicker effect from space current in the
presence of space charge, (3) shot effect from electrons produced by
collision ionization and secondary emission, and (4) space charge
fluctuations due to positive ions. Each of these sources of noise will
be discussed in the following section :

Origin of Noise in Thermionic Amplifier Tubes

Thermal Agitation in the Internal Plate Resistance of the Tube ^
Just as voltage fluctuations are produced by thermal agitation in
resistances comprising the input circuit, so the resistance component
of the impedance between plate and cathode is a source of thermal
noise. This impedance consists of the internal plate impedance of
the vacuum tube in parallel with the external load impedance.
Llewellyn ^ has shown that the resistive component of the internal
plate impedance produces thermal noise as if it were at the temperature
of the cathode. The following formula has been developed by him to
cover the case where the tube impedance and load impedance are
pure resistances:

E? = Akl{r,r,)l{r, + r,y-]{T,r, + Tfr,) f \G,{f)\W (2)

J F

Here rp is the internal plate resistance of the tube, ro the external load
resistance in the plate circuit, G^if) the voltage amplification between
the load resistance ro and the measuring device, and To and T/ re-
spectively the temperatures of the external load resistance and cathode
expressed in degrees Kelvin. The relationship between Gi{f) and
G2(f) is given by

Gi(/) = C?2(/)(/xro)/(ro + r,), (3)

where /j, is the voltage amplification factor of the tube. By assuming
Giij), in equation (2), to be constant over the frequency range F and
substituting for it the value given by equation (3), it is found on
integrating that the thermal noise in the plate circuit of the tube
produces the same effect in the measuring device as a signal applied

^ During the preparation of this paper a paper by E. B. MoulHn and H. D. ElHs
entitled "Spontaneous Background Noise in Amplifiers Due to Thermal Agitation
and Shot Effects" appeared in /. E. E. Jour., 74, 323 (1934). The authors there
contend that no the,rnial noise is produced in the plate impedance of a thermionic
vacuum tube and that shot noise is not altered by the presence of space charge.
With these contentions I cannot agree and I hope to state my definite reasons
therefor at a later date.

^ F. B. Llewellyn, Proc. I. R. £., 18, 243 (1930).

FLUCTUATION NOISE IN VACUUM TUBES 637

to the input circuit whose magnitude at the grid expressed in mean
square volts is given by

V' - ^kT,{r„l^f[_Tfl{T,r,;) + \lr,-]F. (4)

Since the noise of thermal agitation is always present:, this equation
gives the absolute minimum to which fluctuation noise in an amplifying
tube can be reduced after all other causes have been eliminated. It
shows that for the ideal low noise tube in which thermal noise in the
plate circuit is the limiting factor, the noise level may be reduced by a
decrease in the cathode temperature, a decrease in the effective
frequency band, or by an independent decrease in the plate resistance
or increase in the amplification factor. In order to operate at a
minimum noise level the tube should work into a load resistance which
is large in comparison with rpTo/Tp. Under this circuit condition the
noise level is inversely proportional to /j.'^/rp, a quantity often defined
as the '"figure of merit" of an amplifying tube.

Shot Effect and Flicker Effect in the Presence of Space Charge

The theory of the shot effect in the absence of space charge has
been studied quite completely both theoretically and experimentally
by many investigators.^ The results, however, are not applicable
to the study of noise in thermionic vacuum tubes used in high-gain
amplifiers, since a high degree of space charge is required in tubes
used for this purpose. Llewellyn has extended the theory of the shot
effect to cases where partial temperature saturation exists, and ob-
tained a general equation to cover all conditions.^ This equation
reduces to the following form when the load impedance is a pure
resistance:

2ej{dildjy[_rpr,l{rp + r,)J f | G,(f) \ Mf. (5)

Es^ is here the mean square shot voltage across the measuring device,
i the total space current, j the total current emitted by the cathode,
and e the electronic charge (1.59 X 10~^^ coulomb).

A precise experimental verification of this equation is very difficult
because of the difficulty in determining di/dj accurately. Thatcher,*^
however, has made shot measurements in the presence of space charge
(1 ^ di/dj ^ 0.66) which verify the theory within the experimental
error of the determination of dijdj.

^W. Schottky, Ann. d. Physik, 57, 541 (1918); T. C. Fry, Jour. Franklin Inst.,
199, 203 (1925); A. W. Hull and N. H. Williams, Phys. Rev., 25, 147 (1925).
« Everett W. Thatcher, Phys. Rev., 40, 114 (1932).

638

BELL SYSTEM TECHNICAL JOURNAL

Equation (5) shows that as long as space charge is too small to
affect the flow of current, that is when i is equal to j, the mean square
shot voltage is directly proportional to the space current. As emission
is increased, however, space charge begins to control and finally limits
the space current so that the value bi dijdj approaches zero. Thus
the shot voltage increases less rapidly as space charge becomes effective
and then finally decreases rapidly toward zero as complete space
charge control is reached.

Experimental curves showing the effect of space charge on tube
noise are shown in Fig. 1 where abscissae represent space current in

-^,

BARIUM OXIDE

TUNGST

•^

THORIATED TUNGSTEN —5

^-~

—

0 1 2 3 4 5 6 7

SPACE CURRENT IN MILLIAMPERES

Fig. 1 — The effect of space charge on fluctuation noise. Three tubes_having
filaments composed of tungste-Xi, thoriated tungsten, and barium oxide. E''- is the
mean square noise voltage across the output measuring device expressed in arbitrary
units. The variation in space current was obtained by changing the cathode tem-
perature, the plate voltage remaining constant.

FLUCTUATION NOISE IN VACUUM TUBES 639

milliamperes, and ordinates represent mean square noise voltage
across the output measuring device expressed in arbitrary units. The
change in space current was obtained by varying the filament heating
current while the plate voltage remained constant. Tubes having
thoriated tungsten, tungsten, and barium oxide cathodes were used.

At low space currents where no space charge is present the thoriated
tungsten and tungsten filaments each give a pure shot effect, the mean
square voltage increasing linearly with the space current. As the
space current is increased further and space charge sets in, the shot
voltage in each tube goes through a maximum and decreases with
oncoming temperature saturation as suggested by equation (5). With
the approach of complete temperature saturation the noise, however,
does not decrease to zero in accordance with this equation. If it were
possible to reach complete temperature saturation the residual noise
would not be due to the shot effect, but rather to thermal noise in the
plate circuit of the tube, positive ions and secondary emission within
the tube, and other contributing causes. Usually this condition is
approached in the better commercial tubes so that the contribution of
true shot noise is a small part of the total noise.

If the methods used in obtaining equation (4) are applied to equation
(5\ it is found that the shot noise in the plate circuit of the tube
produces the same effect in the output measuring device as a signal
applied to the input circuit whose magnitude at the grid expressed in
mean square volts is

F = 2ej{dildjy{r,l^,fF. (6)

This equation shows that the level of shot noise at the input is lowered
by an increase in the cathode temperature, which increases the degree
of temperature saturation, and by an increase in the ratio yu/r^, which
by definition is the transconductance of the tube, but is independent
of the external load resistance. It should be remembered, however,
that shot noise in the plate circuit should not fix the noise level in
low noise vacuum tubes and that never, as is sometimes done in the
literature, can the noise of an amplifier be calculated as pure shot
noise in the plate circuit, for in the absence of space charge the tube
would not be an amplifier.

Although space charge can counteract the effect of random electron
emission from the cathode so that shot noise is reduced, other factors
can alter the flow of current in such a way that the noise is increased.
This is the case when changes in emission occur over small areas of the
cathode, giving rise to an additional fluctuation which has been
termed flicker effect.^ This type of noise is particularly noticeable

^ J. B. Johnson, Phys. Rev., 26, 71 (1925); VV. Schottky, Phys. Rev., 28, 74.(1926).

640 BELL SYSTEM TECHNICAL JOURNAL

with oxide coated cathodes. Since the flicker effect is due to locaHzed
variations in the emission of the cathode, one would expect it to dis-
appear in the presence of a complete space charge condition.

The experimental curve for the barium oxide coated filament,
Fig. 1, shows a flicker effect many times larger than the shot effect on
which it is superimposed. At low space currents the mean square
flicker effect voltage increases faster than the pure shot noise, a square
law rather than a linear relationship being followed. As space charge
sets in, the flicker effect voltage goes through a maximum and then
decreases with increased space current in the same manner as does
the shot effect voltage. In spite of the large flicker effect, as complete
temperature saturation is approached the total noise is even less than
that found with the thoriated tungsten filament which has no flicker
effect. This illustrates clearly the effectiveness of space charge in
smoothing the space current.

When the control grid of a vacuum tube is floating at its equilibrium
potential, the noise level is much higher than when the grid is con-
nected through an input circuit to the cathode. This increase in
noise is primarily due to thermal noise in the extremely high input
resistance of the tube and to shot noise arising from small grid currents.^
The magnitude of the thermal noise may be calculated, knowing
that the input impedance of the tube consists of its input resistance,
Yg, in parallel with its dynamic grid-to-ground capacitance. In such
a combination the real resistance component, R{j), is related to the
pure resistance, Vg, and the dynamic capacitance, c, according to the
equation

RU) = rgl{\ + ^^''chgT). (7)

According to equation (1) the mean square thermal noise input
voltage is then

V? = UTrg fdflil -{- Air'chgT). (8)

With the grid floating at its equilibrium position (usually slightly
negative with respect to the cathode) the grid current is composed of
two components equal in magnitude but opposite in sign. The one
component consists of electrons reaching the grid, while the other
consists of positive ions reaching and electrons leaving the grid. The
electrons are liberated from the grid by secondary emission, the photo-
electric effect, thermionic emission, and soft X-rays. It should be
pointed out that space charge does not reduce the noise produced by
8 L. R. Hafstad, Phys. Rev., 44, 201 (1933).

FLUCTUATION NOISE IN VACUUM TUBES 641

the shot effect in any of these currents. The general shot effect
equations ^ show that the magnitude of shot noise from these grid
currents is

F? ^ 2ei,r,' f df/{l + 47^Vr//2)^
where ig is the sum of the grid currents regardless of sign.

(9)

Noise Produced by Secondary Effects

In this classification are grouped several sources of disturbance
whose individual effects are very difficult to calculate and measure
under the operating conditions of the vacuum tube. For this reason
the following discussion will include only a general consideration of
the more obvious contributing causes.

Although the cathode is the principal source of electrons which
reach the plate, in actual practice electrons are produced by ionization
of the gas molecules within the tube or by secondary emission resulting
from bombardment of the tube elements. Electrons produced in
this manner are drawn to the plate and generate noise which is not
much affected by the space charge. Assuming a reasonable magnitude
for the current produced in this manner it can be shown by the shot
equations that noise from this source is usually negligible. In cases
where the gas pressure within a tube is above normal, or in screen-
grid and multi-grid tubes having high plate resistances and consider-
able secondary emission, the shot noise from secondary and ionization
electrons may be of the same order of magnitude as thermal noise in
the plate circuit.

Positive ions formed from ionized gas molecules or emitted from
the tube elements are much more effective in producing noise since,
instead of being drawn oft' to the plate, they are attracted into the
space charge region where small disturbances in equilibrium produce
large momentary fluctuations in space current. Due to their large
mass the motions of the ions are relatively slow, so that they are very
effective in this respect. This type of noise is quite disturbing in
amplifying tubes for it tends to become a maximum at complete
temperature saturation. This is illustrated very clearly in the noise
measurements on the tungsten filament shown in Fig. 1. Here positive
ions from the filament begin to show their effect as space charge sets
in, the number of ions and the amount of noise increasing as tempera-
ture saturation is approached. As heard in the loud speaker, this
noise consists of sharp crackling sounds which can easily be dis-
'•* E.g. Ref. 4 or 5.

642 BELL SYSTEM TECHNICAL JOURNAL

tinguished from the steady rustling noise of the shot and thermal
effects.

Ballantine ^^ has recently made calculations and measurements on
the noise due to positive ions from collision ionization in which he
has shown that the mean square noise voltage is roughly proportional
to the gas pressure within the tube and to the 3/2 power of the plate
current. Comparing his results with equation (2), it appears that
under ordinary working conditions the noise due to collision ionization
in a vacuum tube may be of the same order of magnitude as noise
from thermal agitation in its plate circuit. The noise level of tubes
having a poor vacuum, however, may be much higher.

Measurement of Tube Noise

The performance, as regards freedom from noise, of a vacuum tube
used in an amplifier may be indicated by a comparison between the
noise and a signal applied to the grid. Usually we say that the noise
is equivalent to a signal which gives the same power dissipation in the
output measuring instrument as the noise, the frequency of the signal
being suitably chosen with respect to the frequency characteristics of
the amplifier. Since tube noise is distributed over all frequencies and
the noise power increases with the effective band width, it will be
advantageous to express this input signal in equivalent mean square
volts per unit frequency band width, effective over a given frequency
range.

From these considerations it can be seen that the most convenient
standard signal for measuring the equivalent input noise over any
given frequency range is one in which the mean square signal voltage
is distributed equally over all frequencies. With such a signal the
equivalent input noise over any frequency range can be measured
directly, while if an oscillator is used a number of measurements are
required and the result must be computed by graphical integration.
A signal which meets these frequency requirements perfectly is the
noise of thermal agitation. Accordingly, in the measurements to be
described here the standard input signal will be the thermal agitation
voltage of a resistance R, connected between the control grid and
cathode of the tube under test.^

The thermal noise voltage of the grid circuit, referred to the output
measuring device, is given by equation (1), where R{f) is the real
resistance component of an input impedance consisting of the pure
resistance R in parallel with its shunt capacity and that of its leads

10 Stuart Ballantine, Physics, 4, 294 (1933).

FLUCTUATION NOISE IN VACUUM TUBES 643

and of the vacuum tube. In such a combination R{j) is related to
the pure resistance R and the total capacitance c according to equation
(7). In all the measurements described here the factor Air'^c^R'^P is so
small in comparison with unity that it may be neglected without
appreciable error. Under these conditions equation (1) reduces to

AkTR f \G,(f)\'df, (10)

where R is the direct current value of the resistance between control
grid and cathode of the tube under test.

The voltage fluctuations arising from conditions within the tube
produce a mean square voltage output En"^ according to the equation

E?^ f\Vif)\'\G,(f)\Hf, (11)

where | V{f) | ^ is the tube noise at the frequency / for unit frequency
band width, expressed in volts squared and referred to the input
circuit. Letting Vf"^ be the effective value of | V{f)\^ over the band
width of the amplifier we obtain

U If

Gr{f)M. (12)

Since the integrals in equations (10) and (12) are identical it is found
on dividing one equation by the other and solving for Ff^ that :

TV = 4:kTR{E?/E?). (13)

Equation (13) enables one to calculate the magnitude of tube noise
in the frequency range F, per unit cycle band width, in terms of the
thermal noise generated in a resistance R placed in the input circuit. ^^
Since this equation contains no integral the measurements are sim-
plified in that neither standard signal generator nor calibrated amplifier
is required.

Apparatus

The experimental arrangement used in the measurements to be
reported here is given in schematic form in Fig. 2. The system in-
cludes the tube under test, a high gain amplifier, appropriate filters,
an attenuator, and an output measuring device.

" It is assumed that tube noise does not vary with frecjuency, or that the hand
width of the amplifier is so narrow that no appreciable error is introduced in applying
the result.

644 BELL SYSTEM TECHNICAL JOURNAL

TUBE UNDER TEST

Fig. 2 — Schematic amplifier circuit for measuring fluctuation noise in vacuum tubes.

The input circuit consists of the tube under test together with the
variable grid resistor, external load resistor, and batteries for furnishing
the required filament, grid, and plate voltages. Because of the high
value of amplification required and the wide frequency range covered
by the amplifier, this circuit required shielding from external dis-
turbances arising from electrical, mechanical, and acoustical shock.
Accordingly the tube under test was suspended by means of rubber
bands, the whole circuit with the exception of batteries placed inside
a tightly sealed lead lined box, and this box in turn suspended by
means of a system of damped springs. The box with its cover re-
moved and the tube in place is shown in Fig. 3. This shielding was
sufficient to reduce the noise from outside disturbances to such a low
level that no correction had to be made for it at any time.

The high gain amplifier ^^ consists of two separate resistance coupled
units each containing three stages. Each unit is so designed and
shielded that the effect of external disturbances is eliminated. The
total gain obtainable is about 165 db (constant to within 2 db from
10 cycles to 15,000 cycles). Since this gain is in excess of that required
for the study of thermal and tube noises, an attenuator having a
range of 63 db was inserted between the two units. In order to limit
amplification to certain desired frequency bands, specially designed
electric filters were inserted between the first amplifier unit and the
attenuator. Three such filters were used of which one is a low -pass
filter with cut-off around 205 cycles, and the other two are band-pass
filters with mid-frequencies at 1750 and 11,000 cycles respectively.
The frequency characteristic of the amplifier with no filter and with
each filter inserted is shown in Fig. 4.

The recording instrument is a 600-ohm vacuum thermocouple and
microammeter. Conveniently, the deflection of the microammeter is
closely proportional to the mean square voltage applied to the couple.
The procedure in making a measurement of tube noise is as follows:

1- The essential parts of this amplifier were designed by Mr. E. T. Burton.

FLUCTUATION NOISE IN VACUUM TUBES

645

Fig. 3 — Tube under test mounted in the shielding box

;70

m 150

130

120

.'^

^-'

.•'-

100 1000 10,000

FREQUENCY IN CYCLES PER SECOND

SO.OOO

Fig. 4 — Frequency characteristic of amplifier circuit. Curve A with no filter, Curve
B with low pass filter, and Curves C and D with band pass filters.

646 BELL SYSTEM TECHNICAL JOURNAL

With the tube under test operating at zero grid resistance, the attenu-
ator is adjusted to give a convenient deflection of the microammeter
(due to noise in the tube under test). Grid resistance is now added
until this deflection is exactly doubled, thus making En"^ equal to Er^.
This value of input resistance, designated by Rq, is a measure of the
inherent noise of the tube. Substituting Rg in equation (13) the
tube noise is calculated from the relation

F/ = A^kTRo = 1.64 X IQ-^'Rg volt^, (14)

where Rg is expressed in ohms and T is 300° K. (approximate room
temperature).

Noise in Certain Vacuum Tubes ^'

Quantitative measurements of tube noise were made on four different
types of standard Western Electric vacuum tubes, namely: Nos. 102G,
264B, 262A and 259B. These tubes have as low a noise as any tube
obtainable at the present time.

In order to obtain the best signal to noise ratios it was found that
operating conditions different from those normally recommended must
be used. In general, the cathode must be operated at as high a
temperature as possible without impairing the life of the tube, the
negative bias of the control grid must be reduced to as near zero as
possible without causing excessive grid current, and the plate voltage
must be reduced below the value normally recommended. In all
the measurements described here the tube under test was coupled to
the first amplifier unit through a 50,000-ohm load resistance. It was
found that the signal-to-noise ratio could be improved a fraction of a
db by increasing the load resistance (in accordance with equation (4)) ;
this, however, necessitated a large plate voltage which was incon-
venient. Six tubes of each type were tested and the noise data given
below were obtained by averaging the six measurements for each
type. Individual tubes may differ from these average values by as
much as ± 1 db.

No. 102G Tube

This is a three-element, filament-type tube. Its long life, exception-
ally high stability of operation, and good temperature saturation
make it a desirable tube to use in the input stage of certain high-gain
amplifiers. This tube also has a comparatively small microphonic
response to mechanical and acoustical shock although it is not as
good as the No. 262A and the No. 264B tubes in this respect. '^^

" Noise in other types of vacuum tubes has been reported by G. F. Metcalf and
T. M. Dickinson, Physics, 3, 11 (1932); E. A. Johnson and C. Neitzert, Rev. Sc. Inst.,
5, 196 (1934); E. B. Moullin and H. D. M. ElHs, /. E. E. Jour., 74, 323 (1934); W.
Brentzinger and H. Viehmann, Arch. f. Hochfr. und Elektroauk, 39, 199 (1932).

'■' The microphonic response of several types of Western Electric vacuum tubes to
mechanical agitation is reported by D. B. Penick in this issue of the Bell Sys. Tech. Jour.

FLUCTUATION NOISE IN VACUUM TUBES

647

The conditions found most suitable for quiet operation of the No.
102G tube and the corresponding average tube characteristics are
given in the first two columns of Table I. Under these conditions the

TABLE I

Wfstern Electric No. 102G Tube

Tube Characteristics

Noise Data

Operating Conditions

Frequency Range
Cycles per Sec.

P>2
Volt2

Olims

Filament

Voltage, 2.0 volts
Current, 1.0 ampere

Grid

Voltage, — 0.5 volt

Plate
Voltage, 130 volts
Current, 1.2 milli-
amperes

Load

Resistance, 50,000
ohms

Type of Tube, 3 Ele-
ment

Type of Cathode, Ox-
ide Coated Fila-
ment

Amplification Factor,
30.....

Plate Resistance,
45,000 ohms. .

Approx. Dynamic In-
put Capacitance,
80 MMf

10-15,000
5-205
1,750-1,850

10,000-12,000

0.64 X 10-16

2.2
0.58

0.54

3,900

13,600

3,550

3,300

average equivalent tube noise voltage, referred to the grid circuit, is
given in the last column of the same table. These noise data are given
in terms of Rg, the experimentally determined equivalent noise re-
sistance of the tube, and in terms of Vf^, calculated by means of
equation (14), for each of the four frequency ranges shown in Fig. 4.

The No. 102G has the lowest noise of all the tubes tested and was
found suitable for use in the first stage of high-gain amplifiers where
tube noise is the limiting factor, provided it is not required that the
input capacitance and microphonic response to mechanical and
acoustical shock be extremely low.

No. 264B Tube

This is a three-element filament-type tube. Due to the rigid con-
struction and the short filament which is designed to reduce vibration
to a minimum, the microphonic response of the tube to mechanical
and acoustical shock is exceptionally low.^* The extensive system of
spring suspensions and the heavy sound-proof chamber usually re-
quired for shielding low noise tubes may be simplified when using the
No. 264B. In addition, this tube has good temperature saturation,
low power consumption, and high stability of operation.

The operating conditions and noise data for this tube are given in
Table II. Although the noise of this tube is slightly higher than that

15 M. J. Kelly, 5. M. P. E. Jour., 18, 761 (1932).

648

BELL SYSTEM TECHNICAL JOURNAL

TABLE II
Western Electric No. 264B Tube

Tube Characteristics

Noise Data

Operating Conditions

Frequency Range
Cycles per Sec.

I'>2
Volt2

Rg
Ohms

Filament

Voltage, L5 volts
Current, .30 am-
pere

Grid

Type of Tube, 3 Ele-
ment

Type of Cathode, Ox-
ide Coated Fila-
ment

10-15,000
5-205
1,750-1,850

10,000-12,000

1.3 X 10-is

6.6

1.1

1.0

7 650

Voltage, — 0.5 volt
Plate

Voltage, 26 volts

Current, 0.6 milli-
ampere
Load Resistance,

50,000 ohms

Amplification Factor,
7

Plate Resistance,
18,500 ohms

Approx. Dynamic In-
put Capacitance,
30 MAif

40,000
6,800

6,200

of the No. 102G, the lower microphonic response and the lower power
consumption make it a more desirable tube to use in input stages of
certain high gain amplifiers.

No. 262A Tube

This is a three-element tube having an indirectly heated cathode.
It is designed to give a microphonic response to mechanical and
acoustical shock ^^ still lower than that of the 264B. Except for
frequencies below 200 cycles per second it was found that no acoustic
shield was necessary for this tube even when working at extremely low
levels. Although this tube is designed to have a low hum disturbance
resulting from alternating current for heating the cathode (the inter-
ference from this effect can be held to less than 7 X 10~^ equivalent
input volt), direct current power was used in the measurements here
described.

The operating conditions and noise data for the No. 262A tube are
given in Table III.

No. 259B Tube

This is a four-element, screen-grid tube having an indirectly heated
cathode. Its comparatively high amplification factor makes possible
a relatively large gain per stage so that when it is used in the first
stage of a high-gain amplifier succeeding stages contribute nothing to
the total noise.

Noise measurements on the No. 259B tube show that the signal-to-
noise ratio is approximately independent of the plate voltage over a

FLUCTUATION NOISE IN VACUUM TUBES

649

TABLE III
Western Electric No. 262A Tube

Tube Cliaracteristics

Noise Data

Operating Conditions

Frequencv Range

Vf"-

Cycles per Sec.

Volf-

Ohms

Heater

Type of Tube, 3 Ele-

Voltage, 10 volts

ment

Current, 0.32 am-

Type of Cathode, Ox-

pere

ide Coated, Indi-

Grid

rectly Heated

10-15,000

1.3 X 10->e

7,700

Voltage, — 1.0 volt

Amplification Factor,

Plate

15.7

5-205

17.

100,000

Voltage, 44 volts

Plate Resistance,

Current, 1.0 milli-

22,000 ohms

1,750-1,850

1.0

6,400

ampere

Approx. Dynamic In-

Load Resistance,

put Capacitance,

50,000 ohms

2i IXfxi

10,000-12,000

0.84

5,100

wide operating range, but is closely dependent on the plate current as
affected by the control and screen grid voltages. Table IV contains
the operating conditions and noise data for this tube.

Noise measurements were also made on the No. 259B tube with its
control grid floating at equilibrium potential. Using the operating
voltages specified above, the noise level was about 20 db higher than
those given in Table IV. The level can be greatly reduced by oper-
ating the tube at a lower cathode temperature and with lower screen

TABLE IV
Western Electric No. 259B Tube

Tube Characteristics

Noise Data

Operating Conditions

Frequency Range
Cycles per Sec.

Vf-
Volt=

Ohms

Heater

Voltage, 2.0 volts
Current, 1.7 am-
peres
Grid

Control Voltage,
— 1.5 volts

Type of Tube, 4 Ele-
ment Screen Grid

Type of Cathode, Ox-
ide Coated, Indi-
rectly Heated

Amplification Factor,
1,500

10-15,000
5-205
1,750-1,850

10,000-12,000

3.2 X 10-i«

7.7
2.8

2.8

19,800
47,000

Screen Voltage,
22.5 volts
Plate

Voltage, 100 volts
Current 0 6 milli-

Plate Resistance,
2.75 megohms

Approx. Dynamic In-
put Capacitance,
6 0 MMf

17,100
17,000

ampere

Load Resistance,

50,000 ohms

650 BELL SYSTEM TECHNICAL JOURNAL

and plate voltages.^® This reduction in noise is due to a decrease in
current to the floating grid. Using a heater current of 1.3 amperes,
a plate current of 0.1 milliampere, a screen potential of 16.5 volts and
a plate potential of 30 volts the equivalent input noise was 1.4 X 10~^
volt for the entire frequency range from 10 cycles to 15,000 cycles.
Under these operating conditions the floating grid potential was 1.0
volt negative with respect to the cathode, the input resistance 1.4
X 10^° ohms, the dynamic grid-to-cathode capacitance 6 X 10~^^
farad, and each component of grid current about 4.5 X 10~^^ ampere.

Discussion of Results

From the noise data in the preceding tables one can estimate quite
accurately the equivalent input noise voltage of each of the four types
of tubes at any frequency between 5 and 15,000 cycles, and for any
band width within these limits. For example, using the noise data
given in Table I the equivalent input noise voltage of the No. 102G
tube working over a band having sharp cut-offs at 5 cycles and 205
cycles is computed to be

(P)i/2 = (7//r)i/2 = 2.1 X 10-7 volt. (15)

For a band width of 200 cycles with mid-frequency at 10,000 cycles
this noise is reduced to 1.0 X 10~'' volt. It can be seen that for each
type of tube the noise voltage over equal band widths is between 1.5
and 4.5 times greater at frequencies below 200 cycles than at the higher
frequencies.''^

Even at high frequencies the noise voltage is above that expected
from thermal noise in the plate circuit which, as stated above, is the
absolute minimum to which fluctuation noise in a thermionic vacuum
tube may be reduced after all other causes are eliminated. In the
case of the No. 102G tubes for instance, using the operating conditions
of Table I, and assuming 1100° K. as the temperature of the barium
oxide filament, it is found by means of equation (4) that the equivalent
input noise voltage produced by thermal agitation in the plate circuit
is 2.7 X 10-8 volt for a band width of 200 cycles. The total input
noise voltage obtained experimentally at the higher frequencies is
greater than this by a factor of 3.8. In like manner it is found that
the total input noise voltages found experimentally for the Nos. 264B,
262A and 259B tubes are greater than the equivalent input thermal

^® I am indebted to Dr. J. R. Dunning of Columbia University for pointing out this
fact.

1^ Other investigators have also found an increase in tube noise energy at the
lower frequencies. G. F. Metcalf and T. M. Dickinson, Physics, 3, 11 (1932).

FLUCTUATION NOISE IN VACUUM TUBES 651

noise voltages produced in the plate circuit by factors 2.1, 3.7, and 16
respectively. These calculations show that each of these four types
of tubes approaches the requirements of an ideal low noise amplifying
tube although none of them is perfect in this respect.

As stated above, the best signal-to-noise ratio in a high-gain amplifier
is obtained when thermal agitation in the input resistance is responsible
for most of the noise in the amplifier. This condition is met when
the resistance of the input circuit is higher than the value oi Rg for
the input tube. In case the resistance in the input circuit is less than
Rg the input signal and the thermal noise from the input circuit can
be raised above the noise of the tube by using an input transformer
having a sufficiently high voltage step-up. The voltage ratio of the
transformer, and in turn the possible ratio of input circuit thermal
noise to tube noise, is limited, especially at the higher frequencies, by
the dynamic grid-to-ground capacitance of the input tube and its
leads. In such a circuit the No. 259B tube with its lower inter-
electrode capacities and higher tube noise is often more desirable than
even the quietest three-element tubes.

In those high-gain amplifiers in which unavoidably the resistance of
the input circuit is low, the tube rather than thermal agitation in the
grid circuit is responsible for most of the noise. Here the best signal-
to-noise ratio can be obtained by choosing a tube for the initial stage
having the lowest possible noise level. The above measurements
show that one of the three-element tubes, particularly the No. 102G
tube if sufficient shielding is used, is best suited for this purpose.

The lower limits of noise obtainable with high gain amplifiers may
be estimated by means of Fig. 5, which shows the noise as a function
of input resistance and frequency band width when thermal agitation
in the input circuit is responsible for all the noise. The data for this
figure are obtained from the thermal noise relationship

V? = 1.64 X 10-^'RF voh\ (16)

R is expressed in ohms and the temperature has been taken at 300° K.,
which is approximately room temperature. It must be remembered
that the attainment of these noise levels at low input resistances is
limited by the input transformer.

The results of the noise measurements on the No. 259B tube with
floating grid may be compared with the value predicted by equations
(8) and (9). Inserting the tube characteristics obtained by experi-
ment (rg = 1.4 X 10^" ohms, ig = 9 X 10~^^ ampere, and c = 6
X 10~^^ farad), and integrating between the frequency limits 10 cycles

652

BELL SYSTEM TECHNICAL JOURNAL

uli to

10'

-5

f2-f,

CYCLES
" PER SEC.

- \C

"fl

-y'

,--

^ri

-.^

,oP^

• '

_^y^

,yy'

-8

,y^

- vC

^^0^

-9

10*= 10-^ 10*

INPUT RESISTANCE IN OHMS

10-

10°

Fig. 5 — Thermal noise level as a function of input resistance and frequency range.

and 15,000 cycles, it is found that the equivalent thermal noise input
is 0.9 X 10-^ volt, while the shot noise input is 1.4 X 10"^ volt. The
total noise is the square root of the sum of the squares of these values
or 1.7 X 10~^ volt. This agrees with the measured value of 1,4
X 10~^ volt within an error of 20 per cent, which is as accurate as the
determination of the grid currents. These equations may also be
used to calculate the noise originating in the grid circuit when external
resistance or capacitance is connected between grid and cathode, pro-
vided Yg and c are now calculated from the internal and external
impedances in parallel.

A common method of detecting corpuscular or electromagnetic
radiation makes use of an ionization chamber and linear amplifier.
In this circuit the control grid in the first tube of the amplifier is con-
nected to the collecting electrode of the ionization chamber and both
allowed to float at equilibrium potential.^* The shot and thermal
noise in this grid circuit sets a limit to the measurement of extremely
weak radiation. Knowing the value of input capacitance, input

18 H. Greinachcr, Zeits. f. Physik, 36, 364 (1926).

FLUCTUATION NOISE IN VACUUM TUBES 653

resistance, floating grid current, and the frequency limits of the
ampUfier, equations (8) and (9) may be used to calculate this limiting
noise level. For example, if one uses a No. 259B tube with the
operating voltages specified for floating grid, an ionization chamber
having a capacitance of 15 X 10~^^ farad, and an amplifier having a
frequency range from 200 to 5000 cycles per second the limiting
noise level is 1 X 10~^ root mean square volt.

The limiting noise level in a system consisting of a photoelectric
cell and thermionic amplifier is determined by thermal agitation in
the coupling circuit between the photoelectric cell and amplifier, and by
shot noise in the photoelectric current (in circuits where the photo-
electric current is very small and the coupling resistance is very high,
shot noise from grid current in the vacuum tube becomes appreciable).
The noise of thermal agitation may be calculated by means of equation
(8) provided rg is now replaced by R, the coupling resistance. If
vacuum cells are used, the photoelectric current produces a pure shot
noise which can be calculated by equation (9) provided ig is replaced
by /, the photoelectric current. In gas filled photocells where collision
ionization occurs, the noise is in excess of the value calculated in this
manner. ^^ The relative magnitude of shot noise and thermal noise
depends on the values of / and R, and by combining equations (8)
and (9) it is found that

F?/tV = elR/IkT = 19AIR, (17)

where / is expressed in amperes, R in ohms, and T is 300° K. Thus
an increase in either I or R will tend to make shot noise exceed thermal
noise. This is the desirable condition since it furnishes the largest
ratio of signal-to-noise for a given light signal on the photoelectric cell.

In conclusion I wish to acknowledge my indebtedness to Dr. J. B.
Johnson for the helpful criticism he has given during the course of
this work.

13 B. A. Kingsbury, Phys. Rev., 38, 1458 (1931).

Systems for Wide-Band Transmission Over
Coaxial Lines

By L. ESPENSCHIED and M. E. STRIEBY

In this paper systems are described whereby frequency band widths of
the order of 1000 kc. or more may be transmitted for long distances over
coaxial lines and utilized for purposes of multiplex telephony or television.
A coaxial line is a metal tube surrounding a central conductor and separated
from it by insulating supports.

TT appears from recent development work that under some condi-
-*- tions it will be economically advantageous to make use of consider-
ably wider frequency ranges for telephone and telegraph transmission
than are now in use ^' ^ or than are covered in the recent paper on carrier
in cable. ^ Furthermore, the possibilities of television have come into
active consideration and it is realized that a band of the order of one
million cycles or more in width would be essential for television of
reasonably high definition if that art were to come into practical
use.^'^

This paper describes certain apparatus and structures which have
been developed to employ such wide frequency ranges. The future
commercial application of these systems will depend upon a great
many factors, including the demand for additional large groups of
communication facilities or of facilities for television. Their prac-
tical introduction is, therefore, not immediately contemplated and, in
any event, will necessarily be a very gradual process.

Types of High-Frequency Circuits

The existing types of wire circuits can be worked to frequencies of
tens of thousands of cycles, as is evidenced by the widespread applica-
tion of carrier systems to the open-wire telephone plant and by the
development of carrier systems for telephone cable circuits.^- ^ Fur-
ther development may lead to the operation of still higher frequencies
over the existing types of plant. However, for protection against
external interference these circuits rely upon balance, and as the
frequency band is widened, it becomes more and more difficult to
maintain a sufficiently high degree of balance. The balance require-
ments may be made less severe by using an individual shield around

* For references, see end of paper.

654

* Published in Electrical Engineering, October, 1934. Scheduled for presentation
at Winter Convention of A. I. E. E., New York, N. Y., January, 1935.

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 655

each circuit, and with sufificient shielding balance may be entirely
dispensed with.

A form of circuit which differs from existing types in that it is un-
balanced (one of the conductors being grounded), is the coaxial or
concentric circuit. This consists essentially of an outer conducting
tube which envelops a centrally-disposed conductor. The high-
frequency transmission circuit is formed between the inner surface of
the outer conductor and the outer surface of the inner conductor.
Unduly large losses at the higher frequencies are prevented by the
nature of the construction, the inner conductor being so supported
within the tube that the intervening dielectric is largely gaseous, the
separation between the conductors being substantial, and the outer
conductor presenting a relatively large surface. By virtue of skin
effect, the outer tube serves both as a conductor and a shield, the
desired currents concentrating on its inner surface and the undesired
interfering currents on the outer surface. Thus, the same skin effect
which increases the losses within the conductors provides the shield-
ing which protects the transmission path from outside influences, this
protection being more effective the higher the frequency.

The system which this paper outlines has been based primarily upon
the use of the coaxial line. The repeater and terminal apparatus
described, however, are generally applicable to any type of line, either
balanced or unbalanced, which is capable of transmitting the frequency
range desired.

The Coaxial System

A general picture of the type of wide band transmission system which
is to be discussed is briefly as follows: A coaxial line about 1/2 inch in
outside diameter is used to transmit a frequency band of about
1,000,000 cycles, with repeaters capable of handling the entire band
placed at intervals of about 10 miles. Terminal apparatus may be
provided which will enable this band either to be subdivided into more
than 200 telephone circuits or to be used en bloc for television.

Such a wide-band system is illustrated in Fig. 1. It is shown to
comprise several portions, namely, the line sections, the repeaters, and
the terminal apparatus, the latter being indicated in this case as for
multiplex telephony. Two-way operation is secured by using two
lines, one for either direction. It would be possible, however, to
divide the frequency band and use dift'erent parts for transmission in
opposite directions.

A form of flexible line which has been found convenient in the ex-
perimental work is illustrated in Fig. 2 and will be described more fully

656

BELL SYSTEM TECHNICAL JOURNAL

TERMINAL

MULTIPLEXING

APPARATUS

TERMINAL

MULTIPLEXING

APPARATUS

LINES AND REPEATERS

"-^

L^--

Fig. 1 — Diagram of coaxial system.

subsequently. Such a coaxial line can be constructed to have the
same degree of mechanical flexibility as the familiar telephone cable.
While this line has a relatively high loss at high frequencies, the trans-
mission path is particularly well adapted to the frequent application
of repeaters, since the shielding permits the transmission currents to
fall to low power levels at the high frequencies.

Of no little importance also is the fact that the attenuation-fre-
quency characteristic is smooth throughout the entire band and obeys
a simple law of change withjtemperature. (This is due to the fact that
the dielectric is largely gaseous and that insulation material of good
dielectric properties is employed.) This smooth relation is extremely

Fig. 2 — Small flexible coaxial structure.

helpful in the provision of means in the repeaters for automatically
compensating for the variations which occur in the line attenuation
with changes of temperature. This type of system is featured by
large transmission losses which are offset by large amplification, and it
is necessary that the two effects match each other accurately at all
times throughout the frequency range.

It will be evident that the repeater is of outstanding importance in
this type of system, for it must not only transmit the wide band of
frequencies with a transmission characteristic inverse to that of the
line, with automatic regulation to care for temperature changes, but
must also have sufficient freedom from inter-modulation effects to
permit the use of large numbers of repeaters in tandem without objec-

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 657

tionable interference. Fortunately, recent advances in repeater tech-
nique have made this result possible, as will be appreciated from the
subsequent description.

An interesting characteristic of this type of system is the way in
which the width of the transmitted band is controlled -by the repeater
spacing and line size, as follows:

1. For a given size of conductor and given length of line, the band

width increases nearly as the square of the number of the re-
peater points. Thus, for a coaxial circuit with about .3-inch
inner diameter of outer conductor, a 20-mile repeater spacing
will enable a band up to about 250,000 cycles to be transmitted,
a 10-mile spacing will increase the band to about 1,000,000
cycles, and a 5-mile spacing to about 4,000,000 cycles.

2. For a given repeater spacing, the band width increases approxi-

mately as the square of the conductor diameter. Thus, whereas
a tube of .3-inch inner diameter will transmit a band of about
1,000,000 cycles, .6-inch diameter will transmit about 4,000,000
cycles, while a diameter corresponding to a full-sized telephone
cable might transmit something of the order of 50,000,000
cycles, depending upon the dielectric employed and upon the
ability to provide suitable repeaters.

Earlier Work

It may be of interest to note that as a structure, the coaxial form of
line is old — in fact, classical. During the latter half of the last century
it was the object of theoretical study, in respect to skin effect and other
problems, by some of the most prominent mathematical physicists of
the time. Reference to some of this work is made in a paper by
Schelkunoff, dealing with the theory of the coaxial circuit.*^

On the practical side, it is found on looking back over the art that the
coaxial form of line structure has been used in two rather widely differ-
ent applications: first, as a long line for the transmission of low fre-
quencies, examples of which are usage for submarine cables,^- ^ and for
power distribution purposes, and second as a short-distance, high-
frequency line serving as an antenna lead-in.^' ^^

The coaxial conductor system herein described may be regarded as
an extension of these earlier applications to the long-distance trans-
mission of a very wide range of frequencies suitable for multiplex
telephony or television. ^^ Although dealing with radio frequencies,
this system represents an extreme departure from radio systems in that
a relatively broad band of waves is transmitted, this band being con-

658 BELL SYSTEM TECHNICAL JOURNAL

fined to a small physical channel which is shielded from outside dis-
turbances. The system, in effect, comprehends a frequency spectrum
of its own and shuts it off from its surroundings so that it may be used
again and again in different systems without interference.

This new type of facility has not yet been commercially applied. It
is, in fact, still in the development stage. Sufficient progress has
already been made, however, to give reasonable assurance of a satis-
factory solution of the technical problems involved. This progress
is outlined below under three general headings: (1) the coaxial line and
its transmission properties, (2) the wide band repeaters, and (3) the
terminal apparatus.

The Coaxial Line

An Experimental Verification

One of the first steps taken in the present development was in the
nature of an experimental check of the coaxial conductor line, de-
signed primarily to determine whether the desirable transmission prop-
erties which had been disclosed by a theoretical study could be fully
realized under practical conditions. For this purpose a length of
coaxial structure capable of accurate computation was installed near
Phoenixville, Pa. Figure 3 shows a sketch of the structure used and
gives its dimensions. It comprised a copper tube of 2.5 inches outside
diameter, within which was mounted a smaller tube which, in turn,
contained a small copper wire. Two coaxial circuits of different sizes
were thus made available, one between the outer and the inner tubes,
and the other between the inner tube and the central wire. The
instal ation comprised two 2600-foot lengths of this structure.

The diameters of these coaxial conductors were so chosen as to ob-
tain for each of the two transmission paths a diameter ratio which
approximates the optimum value, as discussed later. The conductors
were separated by small insulators of isolantite. The rigid construc-
tion and the substantial clearances between conductors made it pos-
sible to space the insulators at fairly wide intervals, so that the dielec-
tric between conductors was almost entirely air. The outer conductor
was made gas-tight, and the structure was dried out by circulating
dry nitrogen gas through it. The two triple conductor lines were
suspended on wooden fixtures and the ends brought into a test house,
as shown in Fig. 4.

The attenuation was measured by different methods over the fre-
quency range from about 100 kilocycles to 10,000 kilocycles. In-
vestigation showed that the departures from ideal construction occa-
sioned by the joints, the lack of perfect concentricity, etc., had remark-

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 659

ably little effect on the attenuation. In order to study the effect of
eccentricity upon the attenuation, tests were made in which this effect
was much exaggerated, and the results substantiated theoretical pre-
dictions. The impedance of the circuits was measured over the same
range as the attenuation. A few measurements on a short length
were made at frequencies as high as 20,000 kilocycles.

SPACING OF INSULATORS

LARGE

SIZE :

FEET ON STRAIGHTAWAY

FEET ON CURVES

SMALL

SIZE :

FOOT ON STRAIGHTAWAY

INCHES ON CURVES

Fig. 3 — Structure used in Phoenixville installation.

Measurements were secured of the shielding effect of the outer con-
ductor of the coaxial circuit up to frequencies in the order of 100 to
150 kilocycles, the results agreeing closely with the theoretical values.
Above these frequencies, even with interfering sources much more
powerful than would be encountered in practice, the induced currents
dropped below the level of the noise due to thermal agitation of elec-
tricity in the conductors (resistance noise) and could not be measured.

The preliminary tests at Phoenixville, therefore, demonstrated that

660 BELL SYSTEM TECHNICAL JOURNAL

h -^?!?s<*~ ;. _

iimffiiiSfar'

s^H

Hk|^^^ 1

-fr; -MM^^k

L^mi^mmmmm

i^^^M^^H

BHfep^.

W^f^^^^

^■■(M^M

^a^^^

^S£1^hJH^9hH|

'.fr* J t~~

yjl Tgj

ir ""^m

^^^K'f '■^^^^Pl

^^E

■

■ 1

■1

■b

1^9

^^Hk.

'^^^^^H

^^^^^^^^1

^^^^^Rj^Q^^i

^^^^^H

^^^Kfes -I ..:• 'f'i'-is'f

^|[|

Fig. 4 — Phoenixville installation showing conductors entering test house.

a practical coaxial circuit, with its inevitable mechanical departures
from the ideal, showed transmission properties substantially in agree-
ment with the theoretical predictions.

Small Flexible Structures

Development work on wide-band amplifiers, as discussed later,
indicated the practicability of employing repeaters at fairly close in-
tervals. This pointed toward the desirability of using sizes of coaxial
circuit somewhat smaller than the smaller of those used in the pre-
liminary experiments, and having correspondingly greater attenua-
tion. Furthermore, it was desired to secure flexible structures which
could be handled on reels after the fashion of ordinary cable. Ac-
cordingly, several types of flexible construction, ranging in outer
diameter from about .3 inch to .6 inch, have been experimented with.
Structures were desired which would be mechanically and electrically
satisfactory, and which could be manufactured economically, prefer-
ably with a continuous process of fabrication.

One type of small flexible structure which has been developed is
shown in Fig. 2. The outer conductor is formed of overlapping copper
strips held in place with a binding of iron or brass tape. The insula-
tion consists of a cotton string wound spirally around the inner con-
ductor, which is a solid copper wire. This structure has been made in
several sizes of the order of 1 /2 inch diameter or less. When it is to be
used as an individual cable, the outer conductor is surrounded by a

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 661

lead sheath, as shown, to prevent the entrance of moisture. One or
more of the copper tape structures without individual lead sheath may-
be placed with balanced pairs inside a common cable sheath.

Another flexible structure is shown in Fig. 5. The outer conductor
in this case is a lead sheath which directly surrounds the inner conduc-
tor with its insulation. Since lead is a poorer conductor than copper,
it is necessary to use a somewhat larger diameter with this construction
in order to obtain the same transmission efficiency. Lead is also in-
ferior to copper in its shielding properties and to obtain the same de-
gree of shielding the lead tube of Fig. 5 must be made correspondingly
thicker than is necessary for a copper tube.

The insulation used in the structure shown in Fig. 5 consists of hard

RUBBER WASHER INNER CONDUCTOR

(copper)

LEAD OUTER
CONDUCTOR

Fig. 5 — Coaxial structure with rubber disc insulators.

rubber discs spaced at intervals along the inner wire. Cotton string
or rubber disc insulation may be used with either form of outer tube.
The hard rubber gives somewhat lower attenuation, particularly at the
higher frequencies.

Another simple form of structure employs commercial copper tubing
into which the inner wire with its insulation is pulled. Although this
form does not lend itself readily to a continuous manufacturing process,
it may be advantageous in some cases.

Transmission Characteristics
Attenuation

At high frequencies the attenuation of the coaxial circuit is given
closely by the well-known formula:

where R, L, C and G are the four so-called "primary constants" of the
line, namely, the resistance, inductance, capacitance and conductance

662

BELL SYSTEM TECHNICAL JOURNAL

per unit of length. The first term of (1) represents the losses in the
conductors, while the second term represents those in the dielectric.

When the dielectric losses are small, the attenuation of a coaxial
circuit increases, due to skin effect in the conductors, about in accord-
ance with the square root of the frequency. With a fixed diameter
ratio, the attenuation varies inversely with the diameter of the circuit.
By combining these relations there are obtained the laws of variation
of band width in accordance with the repeater spacing and the size of
circuit, as stated previously.

The attenuation-frequency characteristic of the flexible structure
illustrated in Fig. 2, with about .3 inch diameter, is given in Fig. 6.

5 6

TOTAL
ATTENUATION

CONDUCTANCE
LOSS

-. — 1

—

— "

200 400 600 800 1000 1200 1400 1600 I8(

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 6 — Attenuation of small flexible coaxial structure (Fig. 2).

The figure shows also that the conductance loss due to the insulation
is a small part of the total.

It is interesting to compare the curves of the transmission character-
istics of the coaxial circuit with those of other types of circuits. Figure

WIDE-BAND TRANSMISSION OVER COAXIAL LINES

663

NO. 19 GAUGE
CABLE PAIR

''^NO.16 GAUGE
CABLE PAIR

10
9

III

UJ
Q.

^ -r

^^^-^

0.3 INCH

4
3

COAXIAL^^

OPEN

WIRE

2.5 INCH
COAXIAL \

t .

■

300 400 500 600 700

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 7 — Attenuation frequency characteristics of coaxial and other circuits.

7 shows the high-frequency attenuation of two sizes of coaxial circuit
using copper tube outer conductors, of .3 inch and 2.5 inch inner diame-
ter, and that of cable and open-wire pairs in the same frequency range.

Effect of Eccentricity

The small effect of lack of perfect coaxiality upon the attenuation
of a coaxial circuit is illustrated by the curve of Fig. 8, which shows

664

BELL SYSTEM TECHNICAL JOURNAL

TAGE INCREASE IN ATTENUATION
OVER COAXIAL CASE

z '

O
a.

- °o

2 .04 .06 .08 .10 12 14 16 .18 .2

RATIO OF DISTANCE BETWEEN CONDUCTOR AXES
TO INNER RADIUS OF OUTER CONDUCTOR

Fig. 8 — Increase in attenuation of coaxial circuit due to eccentricity.

attenuation ratios plotted as a function of eccentricity, assuming a
fixed ratio of conductor diameters and substantially air insulation.

Temperature Coefficient
With a coaxial circuit, as with other types of circuits, the tempera-
ture coefficient of resistance decreases as the frequency is increased,
due to the action of skin effect, and approaches a value of one-half the
d.-c. temperature coefficient.^^ Thus, for conductors of copper the
a.-c. coefficient at high frequencies is approximately .002 per degree
Centigrade. When the dielectric losses are small, the temperature
coefficient of attenuation at high frequencies is the same as the tempera-
ture coefficient of resistance.

Diameter Ratio
An interesting condition exists with regard to the relative sizes of
the two conductors. For a given size of outer conductor there is a
unique ratio of inner diameter of outer conductor to outer diameter of
inner conductor which gives a minimum attenuation. At high fre-
quencies, this optimum ratio of diameters (or radii) is practically inde-
pendent of frequency. When the conductivity is the same for both
conductors, and either the dielectric losses are small or the insulation
is distributed so that the dielectric flux follows radial lines, the value
of the optimum diameter ratio is approximately 3.6. When the outer
and inner conductors do not have the same conductivity, the optimum
diameter ratio differs from this value. For a lead outer conductor and
copper inner conductor, for example, the ratio should be about 5.3.

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 665

Stranding
Inasmuch as the resistance of the inner conductor contributes a
large part of the high frequency attenuation of a coaxial circuit, it is
natural to consider the possibihty of reducing this resistance by employ-
ing a conductor composed of insulated strands suitably twisted or
interwoven.^* Experiments along this line showed that this method
is impractical at frequencies above about 500 kilocycles, owing to the
fineness of stranding required.

Characteristic Impedance

The high-frequency characteristic impedance of a coaxial circuit
varies inversely with the square root of the effective dielectric constant,
i.e., the ratio of the actual capacitance to the capacitance that would
be obtained with air insulation. The impedance of a circuit having a
given dielectric constant depends merely upon the ratio of conductor
diameters and not upon the absolute dimensions. For a diameter
ratio of 3.6, the impedance of a coaxial circuit with gaseous insulation
is about 75 ohms.

Velocity of Propagation

For a coaxial circuit with substantially gaseous insulation, the veloc-
ity of propagation at high frequencies approaches the speed of light.
Hence the circuit is capable of providing high velocity telephone chan-
nels with their well-recognized advantages. The fact that the ve-
locity at high frequencies is substantially constant minimizes the
correction required to bring the delay distortion within the limits
required for a high quality television band.

Shielding and Crosstalk

The shielding effect of the outer conductor of a coaxial circuit is
illustrated in Fig. 9, where the transfer impedance between the outer
and inner surfaces of the outer conductor is plotted as a function of
frequency. There will be observed the sharp decrease in inductive
susceptibility as the frequency rises. On this account, the crosstalk
between adjacent coaxial circuits falls off very rapidly with increasing
frequency. The trend is, therefore, markedly different from that for
ordinary non-shielded circuits which rely upon balance to limit the
inductive coupling. As a practical matter, less shielding is ordinarily
required to avoid crosstalk than to avoid external interference.

With suitable design the shielding effect of the outer conductor
renders the coaxial circuit substantially immune to external inter-
ference at frequencies above the lower end of the spectrum. Hence
the signals transmitted over the circuit may be permitted to drop

666

BELL SYSTEM TECHNICAL JOURNAL

~~-

0.3"DIAM. 30- MIL
^ COPPER WALL

2.5"DIAM. 60-MIL
COPPER WALL

uj 10
u

Z a: 20

'^ cc 30

go 40
Q- Z
b°50

§S

^1 60
o <

$S 70

"^y 80

UJ cr

CD D 90

o*"
■^^100

20 50 100 200 500 1000 2000 5000

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 9 — Transfer impedance of coaxial circuit.

down to a level determined largely by the noise due to thermal agita-
tion of electricity in the conductors and tube noise in the associated
amplifiers. It appears uneconomical to make the outer conductor
sufficiently thick to provide adequate shielding for the very low fre-
quencies. Also it seems impractical to design the repeaters to trans-
mit very low frequencies. Hence the best system design appears to
be one in which the lowest five or ten per cent of the frequency range
is not used for signal transmission. The coaxial circuit is, however,
well suited to the transmission of 60-cycle current for operating re-
peaters, a matter which will be referred to later.

Broad-Band Amplifiers

In order to realize the full advantage of broad-band transmission,
the repeater for this type of system should be capable of amplifying
the entire frequency band en bloc. Furthermore, it should be so stable
and free from distortion that a large number of repeaters may be op-
erated in tandem. Although high-gain radio frequency amplifiers are
in everyday use, these are generally arranged to amplify at any one
time only a relatively narrow band of frequencies, a variable tuning
device being provided so that the amplification may be obtained at

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 667

any point in a fairly wide frequency range. The high gain is usually
obtained by presenting a high impedance to the input circuits of the
various tubes through tuning the input and interstage coupling cir-
cuits to approximate anti-resonance.

In amplifying a broad band of frequencies, it is difficult to maintain
a very high impedance facing the grid circuits. The inherent capaci-
tances between the tube elements and in the mounting result in a
rather low impedance shunt which can not be resonated over the de-
sired frequency band. It is, therefore, necessary to use relatively low
impedance coupling circuits and to obtain as high gain as possible
from the tubes themselves. The amount of gain which can be ob-
tained without regeneration depends, of course, upon the type of tube,
the number of amplification stages, the band width, and also upon the
ratio of highest to lowest frequency transmitted.

Repeater Gain

The total net gain desired in a line amplifier is such as to raise the
level of an incoming signal from its minimum permissible value, which
is limited by interference, up to the maximum value which the ampli-
fier can handle.

As pointed out above, the noise in a well shielded system is that due
to resistance noise in the line conductors and tube noise in the ampli-
fiers. In some of the repeaters which have been built, the amplifier
noise has been kept down to about 2 db above resistance noise, corre-
sponding to about 7 X 10""^'' watt per voice channel. In a long line
with many repeaters the noise voltages add at random, or in other
words, the noise powers add directly. Assuming, for example, a line
with 200 repeaters, the noise power at the far end would be 200 times
that for a single repeater section. In general, the line and amplifier
noise will not be objectionable in a long telephone channel if the speech
sideband level at any amplifier input is not permitted to drop more
than about 55 db below the level of the voice frequency band at the
transmitting toll switchboard.

The determination of the volume which a tube can handle in trans-
mitting a wide band of frequencies involves a knowledge of the distri-
bution in time and frequency of the signaling energy and of the require-
ments as to distortion of the various components of the signal. The
distribution of the energy in telephone signals has been the subject of
much study. This distribution is known to vary over very wide limits,
depending upon the voice of the talker and many other factors. It is,
therefore, obvious that the problem of summing up the energy of some
hundreds of simultaneous telephone conversations is a difficult one.

668

BELL SYSTEM TECHNICAL JOURNAL

Enough work has been done, however, to indicate fairly well what the
result of such addition will be.

As to distortion in telephone transmission, the most serious problem
has been to limit the intermodulation between various signals which
are transmitted simultaneously through the repeater and appear as
noise in the telephone channel. The requirement for such noise is
similar to that for line and tube noise, and similarly it will add up in
successive repeater sections for a long line. With present types of
tubes operating with a moderate plate potential, the modulation re-
quirement can be met only at relatively low output levels. To im-
prove this situation and also to obtain advantages in amplifier stabil-
ity, the reversed feedback principle employed for cable carrier ampli-
fiers, as described in a paper by H. S. Black, ^■^ has been extended to
higher frequency ranges. It has been found that amplifiers of this
type having 30 db feedback reduce the distortion to such an extent
that each amplifier of a long system carrying several hundred telephone
channels will handle satisfactorily a channel output signal level about
5 db above that at the input of the toll line.

The maximum gain which can be used in the repeater, therefore, is,
in the illustrative case given above of a long system carrying several
hundred telephone channels, the difference between the minimum and
maximum levels of 55 db below and 5 db above the point of reference,
respectively, or a total gain of 60 db. (With a .3-inch coaxial line of
the type shown in Fig. 2, this corresponds to a repeater spacing of
about 10 miles.) If a repeater is to have 60 db net gain and at the

PRE- INPUT

EQUALIZER TRANSFORMER

INTERSTAGE
^COUPLINGS N^

OUTPUT
TRANSFORMER

Fig. 10 — Circuit of 1000-kilocycle three-stage feedback repeater.

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 669

same time about 30 db feedback, it is obvious that the total forward
gain through the amplifying stages must be about 90 db. The circuit
of an experimental amplifier meeting the gain requirements for a
frequency band from 50 to 1000 kilocycles is shown schematically in
Fig. 10.

Gain- Frequency Characteristic

As pointed out above, the line attenuation is not uniform with fre-
quency. For a repeater section which has a loss of, say, 60 db at
1000 kilocycles, the loss at 50 kilocycles would be only about 15 db.
Such a sloping characteristic can be taken care of either by designing
the repeater to have an equivalent slope in its gain-frequency charac-
teristic or by designing it for constant gain and supplementing it with
an equalizer which gives the desired overall characteristic. Both
methods have been tried out, as well as intermediate ones. Figure 11

LINE —DESIRED CHARACTERISTIC
POINTS — ACTUAL CHARACTERISTIC

NO TEMPERATURE REGULATION

0 100 200 300 400 500 600 700 800 900 1000 1100

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 11— Gain of 1000-kilocycle repeater compared with line characteristic.

illustrates such a sloping characteristic obtained by adjusting the
coupling impedances in a three-tube repeater, designed in this case for
60 db gain at 1000 kilocycles. The accompanying photograph, Fig.
12, gives an idea of the apparatus required in such a repeater, apart
from the power supply equipment.

Regulation j or Temperature Changes
It is necessary that the repeater provide compensation for varia-
tions in the line attenuation due to changes of temperature. In the
case of aerial construction such variations might amount to as much
as 8 per cent in a day or 16 per cent in a year. If the line is under-

670

BELL SYSTEM TECHNICAL JOURNAL

ground the annual variation is only about one-third of the above
value and the changes occur much more slowly. On a transcontinental
line the annual variation might total about 1500 db. Inasmuch as it
is desirable to hold the transmission on a long circuit constant within
about ± 2 db, it is obvious that the regulation problem is a serious one.
In a single repeater section of aerial line the variation might amount
to ± 2.5 db per day or ± 5 db per year. Such variations, if allowed

Fig. 12 — Photograph of 1000-kilocycle repeater.

to accumulate over several repeater sections, will drop the signal down
into the noise or raise it so as to overload the tubes. It is, therefore,
advisable to provide some regulation at every repeater in an aerial
line so as to maintain the transmission levels at approximately their
correct position. For underground installations the regulating mech-
anism may be omitted on two out of every three repeaters.

In choosing a type of regulator system the necessity for avoiding
cumulative errors in the large number of repeater sections has been
borne in mind. In view of the wide band available, a pilot channel
regulator system was naturally suggested. Such a scheme employing
two pilot frequencies has been used experimentally to adjust the gain
characteristic in such a way as to maintain the desired levels through-
out the band. The accuracy with which this has been accomplished
for a single repeater section is illustrated in Fig. 13. Over the entire
band of frequencies and the extreme ranges in temperature which may
be encountered, the desired regulation is obtained within a few tenths
of a db.

WIDE-BAND TRANSMISSION OVER COAXIAL LINES

671

70
60

UJ
CQ

O 40
LU

30
Z
<

o:,^

LINES- DESIRED CHARACTERISTICS
POINTS -ACTUAL CHARACTERISTICS

10
0

(

200 300 400 500 600 700 800 900

FREQUENCY IN KILOCYCLES PER SECONT

1000 1100

Fig. 13 — Temperature regulation — line and repeater characteristics.

Repeater Operation, Power Supply, Housing, Etc.

In view of the large number of repeaters required in a broad-band
transmission system it is essential that the repeater stations be simple
and involve a minimum of maintenance. With the repeater design
as described it is expected that most of the repeaters may be operated
on an unattended basis, requiring maintenance visits at infrequent
intervals.

An important factor in this connection is the possibility of supplying
current to unattended repeaters over the transmission line itself. The
coaxial line is well adapted to transmit 60-cycle current to re-
peaters without extreme losses and without hazard. The repeaters
with regulating arrangements as built experimentally for a million-
cycle system are designed to use 60-cycle current, which in this case
appears to have the usual advantages over d.-c. supply. One repeater
requires a supply of about 150 watts. The number of repeaters which
can be supplied with current transmitted over the line from any one
point depends upon the voltage limitation which may be imposed on
the circuit from considerations of safety.

For a repeater of the type described with current supplied over the
line, only a very modest housing arrangement will be required. For
the great majority of stations, it appears possible to accommodate the
repeaters in weatherproof containers mounted on poles, in small huts,
or in manholes.

672

BELL SYSTEM TECHNICAL JOURNAL

Higher Frequency Repeaters
Most of what has been said above appUes particularly to repeaters
transmitting frequencies up to about 1000 kilocycles. However,
study has been given also to repeaters, both of the feedback and the
non-feedback type, for transmitting higher frequencies. Experimental
repeaters covering the range from 500 to 5000 kilocycles have been built
and tested. These were capable of handling simultaneously the full
complement of over 1000 channels which such a broad band will
permit. The frequency characteristic of one of these repeaters, and the
measured attenuation of a section of line of the type tested at Phoenix-
ville are shown in Fig. 14.

M45

Z 35

yy
yX
yX

MEASURED ATTENUATION
OF 2^2 CONDUCTOR
(PHOENIXVILLE TYPE) N^^

y X
y /

• >

V'^AMPLIFIER
^ GAIN

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

FREQUENCY IN MEGACYCLES PER SECOND

Fig. 14 — Frequency characteristic of coaxial line and 5000-kilocycle repeater.

Terminal Arrangements

In order to utilize a broad band effectively for telephone purposes,
the speech channels must be placed as close together in frequency as
practicable. The factors which limit this spacing are: (1) The width of

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 673

speech band to be transmitted and (2), the sharpness of available
selecting networks.

As to the width of speech band, the present requirement for commer-
cial telephone circuits is an effective transmission band width of at
least 2500 cycles, extending from 250 to 2750 cycles. It has been found
that a band of this width or more may be obtained with channels
spaced at 4000-cycle intervals. Band filters using ordinary electrical
elements are available,^ for selecting such channels in the range from
zero to about 50 kilocycles. Channel selecting filters using quartz
crystal elements ^^' ^^ have been developed in the range from about 30
to 500 kilocycles. The selectivity of a typical filter employing quartz
crystal elements is shown on Fig. 15.

V) 70

O 40

1 DECIE

EL — '^

<- 2850 CYCLE BAND-^y

70 71 72 73 74 75 76 77 78 79

FREQUENCY IN KILOCYCLES PER SECOND

Fig. 15 — Frequency characteristic of quartz crystal channel band filter.

Initial Step of Modulation
The initial modulation (from the voice range) may be carried out in
an ordinary vacuum tube modulator or one of a number of other non-
linear devices. The method chosen for the present experimental work

674

BELL SYSTEM TECHNICAL JOURNAL

employs a single sideband with suppressed carrier, using a copper-oxide
modulator associated with a quartz crystal channel filter. The
terminal apparatus required for two-way transmission over a two-
path circuit is shown diagrammatically on the left-hand side of Fig. 16.

OTHER
TRANSMITTING

CHANNEL

BAND FILTERS

(64-108 KC)

OF THIS GROUP

CHANNEL
MODULATOR

68-72 I--*
KC ^-r

64-68 r-t
KC Y-*-

TELEPHONE
SET ■

HYBRID

■^m fW^- NET-

COIL

60-64
KC

TRANSMITTING

CHANNEL

BAND FILTER

^1^

DEMODULATOR
AMPLIFIER

RECEIVING

CHANNEL

BAND FILTER

60-64
KC

CHANNEL
DEMODULATOR

OTHER

RECEIVING

CHANNEL

BAND FILTERS

(64-108 KC)

OF THIS GROUP

GROUP
MODULATOR

60-108
KC

972-
1020
KC

I f-; 876-
t-^-l924KC

I 1 —.'

I ♦•-I 924- 1
|"}-(972KC|

I I ' "■-

) I

[ I TRANS-

I I MITTING

I 'AMPLIFIER

I I
-4-1.

OTHER

[transmitting

^ GROUP
BAND FILTERS

1080
KC

TRANSMITTING

GROUP
BAND FILTER

RECEIVING

GROUP
BAND FILTER

EAST

COAXIAL LINE
60-1020 KC

60-108
KC

972-
1020
KC

I GROUP
I DEMODULATOR

64-68 I- -♦
KC h-i-

68-72 I--*
KC 1—1-

I RECEIVING
I AMPLIFIER

f-i 972- :

f-t-]924KC;

f-i 924- ;
f-T-|876KC!

WEST

OTHER
RECEIVING
GROUP
BAND FILTERS

Fig. 16— Schematic of four-wire circuit employing two steps of modulation.

A frequency allocation which has been used for experimental pur-
poses employs carriers from 64 to 108 kilocycles for the initial step of
modulation. The lower sidebands are selected and placed side by
side in the range from 60 to 108 kilocycles, as illustrated in Fig. 17,
forming a group of 12 channels.

Double Modulation
In order to extend the frequency range of a system to accommodate
a very large number of channels, it appears to be more economical to
add a second step of modulation rather than carry the individual
channel modulation up to higher frequencies. Such a second step of
modulation has been used experimentally to translate the initial group
of 12 channels en bloc from the range 60 to 108 kilocycles up to higher
frequencies. It is possible to place such groups of channels one above
another as illustrated in the upper part of the diagram of Fig. 18, up

WIDE-BAND TRANSMISSION OVER COAXIAL LINES

675

VOICE
FREQUENCY
CHANNELS

>fVvYyivvvtiv

FREQUENCY IN | KILOCYCLES PER SECOND
48 KC GROUP -

Fig. 17 — Diagram illustrating frequency allocation for first step of modulation.

48 KG
GROUPS

□n

f » '' " ' T

0 60108

780 4 1020 ^ 1260 4 1500 4 1740
FREQUENCY IN KILOCYCLES PER SECOND

I 240 KC 1
GROUPS

Fig. 18 — Diagram illustrating frequency allocation for two or three steps of

modulation.

676 BELL SYSTEM TECHNICAL JOURNAL

to about 1000 kilocycles, wasting no frequency space between groups
and thus keeping the channels spaced at intervals of 4 kilocycles
throughout the entire range.

The apparatus required for this purpose is shown schematically in
Fig. 16, which illustrates the complete terminal arrangements for a
single channel employing double modulation. The figure indicates by
dottled lines where the other channels and groups of channels are con-
nected to the system.

A modulator for shifting the frequency position of a group of chan-
nels inherently yields many different modulation products as a result
of the intermodulation of the signal frequencies with the carrier fre-
quency and/or with one another. Out of these products only the
"group sideband " is desired. The number of the modulation products
resulting merely from the lower ordered terms of the modulator re-
sponse characteristic is extremely large. All such products must be
considered from the standpoint of interference either with the group
which is wanted in the output or with other groups to be transmitted
over the system. Various expedients may be used to avoid inter-
ference as follows: (1) A proper choice of frequency allocation will
place the undesired modulation products in the least objectionable
location with respect to the wanted signal bands; (2) a high ratio of
carrier to signal will minimize all products involving only the signal
frequencies; (3) the use of a balanced modulator will materially reduce
all products involving the second order of the signal ; (4) selectivity in
the group filters will tend to eliminate all products removed some dis-
tance from the wanted signal group. Giving due regard to these
factors, balanced vacuum tube group modulators have been developed
which are satisfactory for the frequency allocations employed.

Triple Modulation
For systems involving frequencies higher than about 1000 kilo-
cycles it may be desirable to introduce a third step of modulation.
In some experiments along this line a "super-group" of 60 channels,
or five 12-channel groups, has been chosen. The lower part of Fig.
18 illustrates, for a triple modulation system, the shifting of super-
groups of 60 channels each to the line frequency position. This
method has been employed experimentally up to about 5,000 kilo-
cycles. It is of interest to note that even in extending these systems
to such high frequencies, channels are placed side by side at intervals
of 4000 cycles to form a practically continuous useful band for trans-
mission over the line.

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 677

Demodulation

On the receiving side the modulation process is reversed. The
apparatus units are similar to those used on the transmitting side, and
are similarly arranged. Figure 16 illustrates this for the case of double
modulation.

Carrier Frequency Supply

In systems operating at higher frequencies it is necessary that the
carrier frequencies be maintained within a few cycles of their theoretical
position in order to avoid beat tones or distortion of the speech band.
Separate oscillators of high stability could, of course, be used for
the carrier supply but it appears more economical to provide carriers
by means of harmonic generation from a fundamental basic frequency.
Such a base frequency may be transmitted from one end of the cir-
cuit to the other, or may be supplied separately at each end.

Television

The broad band made available by the line and repeaters may be
used for the transmission of signals for high-quality television. Such
signals may contain frequency components extending over the entire
range from zero or a very low frequency up to a million or more cycles."'
The amplifying and transmitting of these frequencies, particularly the
lower ones, presents a serious problem. The difficulty can be over-
come by translating the entire band upward in frequency to a range
which can be satisfactorily transmitted. To effect such a shift, the
television band may first be modulated up to a position considerably
higher than its highest frequency and then with a second step of modu-
lation be stepped down to the position desired for line transmission.

This method is illustrated in Fig. 19 for a 500-kc. television signal
band. The original television signal is first modulated with a rela-
tively high frequency, two million cycles in this case (Ci). The lower
sideband, extending to 1500 kilocycles, is selected and is modulated
again with a frequency of 2100 kilocycles (G). The lower sideband
of 100 to 600 kilocycles is selected with a special filter so designed that
the low frequency end is accurately reproduced. The television
signal then occupies the frequency range of 100 to 600 kilocycles as
shown on the diagram and may be transmitted over a coaxial or other
high frequency line. At the receiving end a reverse process is em-
ployed. The same method using correspondingly higher frequencies
may be used for wider bands of television signals.

678

BELL SYSTEM TECHNICAL JOURNAL

FREQUENCY IN KILOCYCLES PER SECOND

2000 KC
CARRIER

PHOTO-
ELECTRIC
CELL

,ST
FILTER

,ST
MODULATOR

COAXIAL
LINE

Fig. 19 — Double modulation method for translating television signals for wire line

transmission.

Other Communication Facilities

The telephone channels provided by the system may be used for
other types of communication services, such as multi-channel tele-
graph, teletype, picture transmission, etc. For the transmission of a
high-quality musical program, which requires a wider band than does
commercial telephony, two or more adjacent telephone channels may
be merged. The adaptability of the broad-band system to different
types of transmission thus will be evident.

As already noted, the commercial application of these systems for
wide-band transmission over coaxial lines must await a demand for
large groups of communication facilities or for television. The re-
sults which have been outlined are based upon development work in
the laboratory and the field, and it is probable that the systems when
used commercially will differ considerably from the arrangements
described.

WIDE-BAND TRANSMISSION OVER COAXIAL LINES 679

References

1. E. H. Colpitts and O. B. Blackwell, "Carrier Current Telephone and Teleg-

raphy," A. I. E. E. Trans., Vol. 40, February 1921, p. 205-300.

2. H. A. Affel, C. S. Demarest, and C. W. Green, "Carrier Systems on Long Dis-

tance Telephone Lines," A. I.E. E. Trans., Vol. 47, October 1928, 1360-1367.
Bell Sys. Tech. Jour., Vol. VH, July 1928, p. 564-629.

3. A. B. Clark and B. W. Kendall, "Communication by Carrier in Cable," Elec.

Engg., Vol. 52, July 1933, p. 477-481, Bell Sys. Tech. Jour., Vol. XII, July 1933,
p. 251-263.

4. P. Mertz and F. Gray, "Theory of Scanning and Its Relation to the Characteris-

tics of the Transmitted Signal in Telephotography and Television," Bell Sys.
Tech. Jour., Vol. XIII, July 1934, p. 464.

5. E. W. Engstrom, "A Study of Television Image Characteristics," Proc. I. R. E.,

Vol. 21, December 1933, p. 1631-1651.

6. S. A. Schelkunoff, "The Electromagnetic Theory of Coaxial Transmission Lines

and Cylindrical Shields." Bell Sys. Tech. Jour., Vol. XIII, October 1934.

7. J. R. Carson and J. J. Gilbert, "Transmission Characteristics of the Submarine

Cable," Jour. Franklin Institute, Vol. 192, December 1921, p. 705-735.

8. W. H. Martin, G. A. Anderegg and B. W. Kendall, "The Key West-Havana

Submarine Telephone Cable System," Trans. A. I. E. E.,Vo\. H, 1922, p. 1-19.

9. British Patent No. 284,005, C. S. Franklin, January 17, 1928.

10. E. J. Sterba and C. B. Feldman, "Transmission Lines for Short-Wave Radio

Systems," I. R. E. Proc, Vol. 20. July 1932, p. 1163-1202; also Bell Sys. Tech.
Jour., Vol. II, July 1932, p. 411^50.

11. U. S. Patents No. 1,835,031, L. Espenschied and H. A. Affel, December 9, 1931,

and No. 1,941,116, M. E. Strieby, December 26, 1933.

12. E. I. Green, "Transmission Characteristics of Open-Wire Lines at Carrier

Frequencies," A. I. E. E. Trans., Vol. 49, October 1930, p. 1524-1535; Bell
Sys. Tech. Jour., Vol. IX, October 1930, p. 730-759.

13. H. A. Affel and E. I. Green, U. S. Patent No. 1,818,027, Aug. 11, 1931.

14. H. S. Black, "Stabilized Feed-Back Amplifiers," Electrical Engineering, Vol. 53,

January 1934, p. 114-120. Bell Sys. Tech. Jour., Vol. XIII, January 1934,
p. 1-18.

15. L. Espenschield, U. S. Patent No. 1,795,204, March 3, 1931.

16. W. P. Mason, "Electrical Wave Filters Employing Quartz Crystals as Elements,"

Bell Sys. Tech. Jour., Vol. XIII, July 1934, p. 405.

Regeneration Theory and Experiment *
By

E. PETERSON, J. G. KREER, AND L. A. WARE

A comprehensive criterion for the stability of linear feed-back circuits
has recently been formulated by H. Nyquist, in terms of the transfer factor
around the feed-back loop. The importance of any such general criterion
lends interest to an experimental verification, with which the paper is
primarily concerned.

The subject is dealt with under five principal headings. The first sec-
tion reviews some of the criteria for oscillation to be found in the literature of
vacuum tube oscillators. The second describes the derivation of Nyquist's
criterion somewhat along the lines followed by Routh in one of his investiga-
tions of the stability of dynamical systems. The third part deals with two
experimental methods used in measuring the transfer factor. The fourth is
concerned with the particular amplifier circuit used in the test of Nyquist's
criterion. The last section applies the criterion to a nonlinear case, and to
circuits including two-terminal negative impedance elements.

IN a comparatively recent paper on "Regeneration Theory," ^ Dr.
Nyquist presented a mathematical investigation of the conditions
under which instability ^ exists in a system made up of a linear ampli-
fier and a transmission path connected between its input and output
circuits. The results of the investigation are of interest because of
their obvious application to amplifiers provided with feed-back paths,^
as well as to the starting conditions in oscillators. As a result of his
general analysis. Dr. Nyquist arrived at a criterion for stability, ex-
pressed in particularly simple and convenient form, which is not re-
stricted in its range of application to particular amplifier and circuit
configurations.

The great value attached to a criterion as precise and as general as
Nyquist's makes it desirable to submit the criterion to an experimental
test. One particularly striking conclusion drawn from this criterion is
that under certain conditions a feed-back amplifier may sing within
certain limits of gain, but either reduction or increase of gain beyond
these limits may stop singing. A feed-back amplifier satisfying these
conditions was set up, and the experimental results were found to be in
agreement with this conclusion.

* Published in Proc. I. R. E., October, 1934.

1 Bell. Sys. Tech. Jour., vol. XI, p. 126.

2 Instability is used in the sense that a small impressed force, which dies out in
course of time, gives rise to a response which does not die out.

^Electrical Engineering, July, 1933; Bell Sys. Tech. Jour., p. 258, July, 1933.

680

REGENERATION THEORY AND EXPERIMENT 681

It is interesting to compare the criterion with those derived for the
mechanical systems of classical dynamics. In his Adams Prize Paper
on "The Stability of Motion," ■* and again in his "Advanced Rigid
Dynamics," ^ Routh investigated the general problem of dynamic
stability and established a number of criteria based upon various
properties of dynamical systems. When applied to the problem of
feed-back amplifiers, keeping Nyquist's result in mind, one of them is
found to be equivalent to Nyquist's criterion, although expressed in
different terms and derived in a different way.

To provide a background for the experiments, we propose to state
some of the criteria for stability which are to be found in the literature
of vacuum tube oscillators, and to compare them with Nyquist's or
Routh's criterion, the development of which is most conveniently de-
scribed somewhat along the lines followed by Routh. Following this
we shall deal with the experimental methods and apparatus which were
used in testing the criterion, and conclude with some extensions of the
criterion.

Circuit Analysis and Stability

Conditions required for the starting of oscillations in linear feed-
back circuits, corresponding to instability, are to be found in the litera-
ture of vacuum tube oscillator circuits, expressed in a number of os-
tensibly different forms. These are usually based upon the familiar
mesh differential equations for the system which involve differentia-
tions and integrations of the mesh amplitudes with respect to time.
Using the symbol p to denote differentiation with respect to time, each
mesh equation becomes formally an algebraic one in p, involving the
circuit constants and the mesh amplitudes. The solution of this sys-
tem of equations is known to be expressible as the sum of steady state
and transient terms. The transient terms are each of the form Bk
eP*', the BkQ being fixed by initial conditions, and the pkS being de-
termined from the circuit equations. If we set up the determinant of
the system of equations — the discriminant — and equate it to zero, the
roots of the resulting equation are the ^^'s above. In general each
mesh equation involves p to the second degree at most, and with n
meshes the discriminant is of degree 2n at most. Accordingly we may
express the determinantal equation as

F{p) = 0= K(P- P,){P - P2) ■■■ (P- p2n). (1)

As for the steady state term, in the simplest case in which a sinu-
soidal wave of frequency wjlir is impressed, it is equal to the impressed

4 Macmillan, 1877.

^ Macmillan, 6th edition, 1905.

682 BELL SYSTEM TECHNICAL JOURNAL

voltage divided by the discriminant and multiplied by the appropriate
minor of the determinant, in which p is replaced byjco. The character
of the response due to a slight disturbance and in the absence of any
periodic force is determined by the exponentials. In general, pk is a
complex quantity which may be written as ak -\- jwk- It is apparent
that in the critical case for which ak is zero, the corresponding term be-
comes e^"*', corresponding to an oscillation invariable in amplitude,
of frequency wkfix. If ak is negative, as is ordinarily the case when
the system is passive (containing no amplifier or negative impedance),
then the oscillation diminishes in course of time. When ak is positive,
however, the oscillation increases with time, and the system is said to
be unstable. Evidently the stability of a system is determined by the
signs of the a^'s.

Several criteria which have previously been enunciated for the
maintenance of free oscillations are deducible from the above. One
states that the discriminant must vanish when p takes on the value jco.
Another states that the damping (a/;) must be zero at the frequency of
oscillation. These are clearly equivalent. Two derived criteria may
also be mentioned, based upon the properties of the system when the
circuit is broken. The first of these states that if the impedance is
measured looking into the two terminals provided by the break, the
impedance must be zero at the frequency of steady oscillation.

The second criterion involving the transfer factor has become fairly
widespread, perhaps because it leads to a simple and plausible physical
picture. To determine the transfer factor around the feed-back loop,
the loop is broken at a convenient point, and the two sets of terminals
formed by the break are each terminated in a passive impedance equal
to that which is connected in the normal (unbroken) condition. Then
when a voltage of frequency co/27r is applied to one of the pairs of ter-
minals so provided — the input terminals ^ — and the corresponding
voltage is measured across the other pair, the transfer factor A (jco) is
obtained as the vector ratio of the output voltage to the input voltage.

The manner in which the transfer factor enters into the problem may
be demonstrated directly by comparing the voltages at any point of the
main amplifier circuit under the two conditions in which the feed-back
path is opened and closed respectively. If with the feed-back path
open the voltage at any such point is Ee''\ then when the feed-back
path is closed the voltage will be changed ^ to

£e^7[l -A{pn

^ Input terminals are those across which an impressed potential leads to propaga-
tion in the normal direction of amplifier transmission.
' Bell Sys. Tech. Jour., Vol. XI, p. 128.

REGENERATION THEORY AND EXPERIMENT

683

This may be shown as follows with reference to the particular circuit
of Fig. 1 : If the feed-back circuit is broken and then properly termi-
nated, the voltage existing across the input is taken as e. Now suppose
the feed-back path to be restored. Designating the voltage existing
across the input in the presence of feed-back as ei we have ei = e -\- Aci,
from which the above equation follows.

AMPLl-

FJ^R

■

A A /. A

*C^

< z

'^ ^

NET-
WORK

rS)E

AMPLI -
FIER

<;^

^ z

NET-
WORK

r\j) E

Fig. 1 — Series type feed-back; loop broken and terminated at left, normal feed-back

circuit at right.

If we let Fi{p) represent the discriminant of the system when the
loop is broken and terminated, then the roots of the equation formed
by setting the discriminant equal to zero are assumed to have positive
real parts. Now for the corresponding discriminant when the loop is
restored, we have in accordance with the above considerations

F{p) = [(1 - A{p):[F,{p).

In setting this discriminant equal to zero to obtain the roots, the
only ones which have nonnegative real parts are those corresponding to
the feed-back term

f{p)=\-A{p). (2)

The above-mentioned criterion may be deduced from this expression.
For steady oscillations to exist the output potential must be identical
in amplitude and in phase with that existing across the input at the fre-
quency of oscillation {p = jco), in which case the transfer factor is unity.
This seems reasonable on the basis that when the input and output
terminals are connected through, the oscillation will neither increase
nor decrease with time. It may be demonstrated by direct analysis
that these several criteria, framed for the critical case of undamped
oscillations, all lead to the same correct conclusion.

Of course in any actual oscillating circuit it is practically impossible
to get these conditions fulfilled exactly, and what is ordinarily done
in the practical design of oscillating circuits is to ensure that the
voltage fed back will be greater than that required to produce oscilla-
tion. This evidently goes a step further than the above criteria, and

684 BELL SYSTEM TECHNICAL JOURNAL

reliance is placed upon the nonlinear properties of the circuit to ful-
fill the criteria automatically. The procedure is known by experiment
to be effective in the usual type of oscillating circuit. In particular
forms of feed-back circuits, however, it may be demonstrated that
the transfer factor may be made greater than unity without giving rise to
oscillations. This situation was investigated experimentally, and found
to be in accord with the stability criterion stated by Nyquist.

Nyquist's Criterion

The explicit solution of (1) for the pkS> demands an exact knowl-
edge of the configuration of the amplifier and feed-back circuits. When
the number of meshes is large, the solution involves much labor. If we
wish simply to observe whether or not the system Is stable, however,
we need not obtain explicit solutions for the roots; in fact, all we need
to know is whether or not any one of the pkS has its real part positive.
It turns out that when we know the transfer factor as a function of fre-
quency, by calculation or by measurement, a simple inspection of the
transfer factor polar diagram suffices for this purpose. This diagram is
constructed by plotting the imaginary part of the transfer factor
against the real part for all frequencies from minus to plus infinity.^

To obtain Nyquist's criterion we consider the vector drawn from
the point (1, 0) to a point moving along the polar diagram; if the net
angle which the vector swings through in traversing the curve is zero,
the system is stable; if not, it is unstable. To express it in the terms
used by Routh, if we set I — A (jco) = P -\- jQ, and observe the changes
of sign which the ratio P/Q makes when P goes through zero as the
frequency steadily increases, the system is stable when there are the
same number of changes from plus to minus as from minus to plus.
It may be demonstrated that these two statements are equivalent.

The way in which the above procedures may be shown to reveal the

existence of a root with positive real part may be outlined somewhat

along the lines followed by Routh in his analysis.^ Since p is a. com-

* The transfer factor for negative frequencies A (— Jco) is the complex conjugate of
that for positive frequencies A{ju). Thus, if

A(jw) = X +jY,
then

Ai-jo.) = X -jY.

^ A number of restrictions on the generality of the analysis may be noted. It is
assumed that A(p) has no purely imaginary roots, although the result in this case is
otherwise evident. Further it is assumed that A{p) goes to zero as \p\ becomes
infinite, and that no negative resistance elements are included in the amplifier.
Another point which should be mentioned is that the analysis does not apply to the
stability in any conjugate paths that may exist. This point may be exemplified by
the balanced tube or push-pull amplifier, in the normal transmission path of which the
tubes of a stage act in series. When the series output is connected back to the series

REGENERATION THEORY AND EXPERIMENT

685

plex quantity in general, any value which it may take is representable
as a point on a plane — the ^-plane of Fig. 2. Since only values of p
with positive real parts concern us, attention may be confined to the
right-hand half of the p-p\ane. Now draw a closed contour C in the
right-hand half of the ^-plane which encloses the root pk. It is evident

Fig. 2 — Plot of two contours C and Ci in the ^-plane. C encloses the root pk
while Ci does not enclose pk. The vector p — pk covers 360 degrees as p traverses C,
and covers the net angle of zero as p traverses Cj.

upon inspection of Fig. 2 that the vector extending from the root pk
to the contour makes a complete revolution (360 degrees) in following
the closed path. If the contour does not enclose the root, however, as
for Ci, then it is clear that when the vector from the root to the contour
traverses the whole contour, the net angle turned through by the vec-
tor is zero. In the region under consideration we may write

KP) - (P- PM{P),

where 0(^) has no zeros within the contour. Hence, when p traverses
a closed path and {p — pk) turns through 360 degrees or through zero
the same angle is covered by f{p). If for some different contour sev-
eral roots are enclosed, it may be shown that f{p) turns through one
complete revolution for each of the enclosed roots when p traverses the
contour.

In the form in which these considerations are stated, they are not
suitable to practical application since complex values of p are in-
volved. Ordinarily, of course, only imaginary values {p = joj) are con-
veniently accessible to us since it is a comparatively simple matter to

input, stability of the resultant loop has in general no bearing upon the stability of
the path formed with the two tubes of each stage in parallel, since the series and shunt
paths are conjugate to one another. To establish the staliility of the shunt or parallel
path, the transfer factor for that path must be separately determined. In general,
the stability criterion applies only to the particular loop invcbligated, and not to any
other existent loop.

686

BELL SYSTEM TECHNICAL JOURNAL

measure the response with a sinusoidal impressed wave, but it would
involve great difficulties of experiment as well as of interpretation to
determine the response with negatively damped waves corresponding
to values of p in the right-hand half of the plane. However, these re-
sults may be brought within the field of practical experience by a pro-
cedure widely used for the purpose.

To include all roots in the right-hand half of the ^-plane, the con-
tour must be taken of infinite extent. The path ordinarily followed for
this purpose extends from the value -\- R to — Ron the imaginary axis,
and is closed by a semicircle of radius R, where R is assumed to expand
without limit. It may be noted that in actual amplifier circuits the
transfer factor becomes zero when \p\ becomes infinite, so that A{p)
is zero along the semicircular part of the closed contour. Conse-
quently, the only values of A (p) which dififer from zero are those corre-
sponding to finite values of p, along the imaginary axis. In other
words, the plot of A(p) under these conditions comes down to the plot
of A (jo) where w is finite. Hence, if we plot A (jco) for all values of co
from minus to plus infinity, there will be no roots with positive real
parts and the system will be stable when the vector from (1, 0) to the
curve sweeps through a net angle of zero. The system will be unstable
when the vector sweeps through 360 degrees, or an integral multiple
thereof.

Two types of transfer factor curves may be considered as illustra-
tions. The first of these shown in Fig. 3 corresponds to that for a re-

3-M-C

Fig. 3 — Schematic of a reversed feed-back oscillator circuit at the left. At the
right plot of the transfer factor A (jco) around the feed-back loop of Fig. 3a over the
frequency range from zero to very high frequencies. The imaginary part of the
transfer factor is plotted as ordinate against the real part as abscissa for the three
curves a, b, c, which correspond to increasing gains around the loop. Condition a
is stable, while b and c are unstable.

REGENERATION THEORY AND EXPERIMENT 687

versed feed-back oscillator circuit, the three curves marked a, b, c, cor-
responding to progressively increasing gains around the loop. It will
be observed that after the maximum gain has reached and exceeded
unity, that the circuit is unstable, since the point (1, 0) is then enclosed.
This state of affairs may be contrasted with that existing in the par-
ticular form of feed-back circuit to which Fig. 4 applies. Again the

Fig. 4 — Transfer factor diagram for a particular form of feed-back circuit, curves
a, 0, c, corresponding to increasing gains around the feed-back loop. Conditions a
and c are stable, b is unstable.

three curves a, b, c, correspond to progressively increasing gains around
the feed-back loop. As the gain is increased the system is first stable
(a), then unstable (&), and finally stable (c), since it is only within
curve (b) that the point (1, 0) is enclosed. This striking example is the
one which was investigated experimentally. The methods used in de-
termining the transfer factor diagram form the subject of the next
section.

Measuring Methods

Application of the Nyquist stability criterion requires the de-
termination of the vector transfer factor around the feed-back loop
at all frequencies. This is usually effected by opening the circuit at any
point which provides convenient impedances looking in both directions
from the break. These points are then connected to an oscillator and
to suitable measuring circuits, which are to be described. Care must
be taken to ensure that the oscillator and measuring circuit impedances
are equal to the output and input impedances respectively of the
circuit under test. This precaution is necessary in order that the trans-
fer factor in the measuring condition may not differ significantly
from that existing in the operating condition.

688 BELL SYSTEM TECHNICAL JOURNAL

Two methods of measurement have been found useful. The first is
a null method capable of good precision over a wide frequency range.
The second is a visual method in which the transfer factor polar dia-
gram is traced on the screen of a cathode ray oscillograph. This
method is not capable of very great precision and, in the model used,
the frequency range is somewhat restricted. However, it permits of a
rapid survey of the situation for which its precision is adequate, before
proceeding with the slower and more precise measurements of the null
method, where the latter are required. By making such a preliminary
survey the critical frequency ranges can be mapped out for precise
measurement, thereby eliminating a large amount of unnecessary labor.

Null Method

In the more precise measurements extending over a wide frequency
range, special care is required to ensure freedom from errors in the
measurement of phase angles and amplitudes. Much of the difficulty
associated with direct measurement over wide frequency ranges is
avoided by the use of a simple demodulation scheme. In this scheme,
the potentials to be compared are modulated down to a fixed frequency
(in actual use 1000 cycles) regardless of the frequency at which the test
is being made. In this way a minimum portion of the circuit carries
the high frequency. Further this permits the use of voice frequency at-
tenuators, phase shifters, and amplifiers which in fact require calibra-
tion at only a single frequency.

In this arrangement, as shown in Fig. 5, demodulators are shunted
across the input and output terminals of the circuit under test. A
single oscillator supplies the carrier to both demodulators, its frequency
differing by 1000 cycles from the frequency supplied to the circuit
under test. The demodulated outputs are connected through attenua-
tors and phase shifters to a common amplifier detector. The attenua-
tors and phase shifters are adjusted until the detector gives a null read-
ing. When this condition obtains the difference in the attenuator set-
tings in the two branches is equal to the gain or loss of the circuit
under test, and the difference in the phase shifter settings is either
equal to or the negative of the phase shift of the circuit under test.
To show this, denote the amplifier output voltage by Po cos {livft — (p),
and the beat frequency voltage supplied to the demodulators by P cos
27r(/ ± 1000)^. The demodulated output, proportional to the product
of the two applied waves, is then

PPoCOS (27r-1000/ T </)).
Correspondingly, the demodulated output from the other demodulator

REGENERATION THEORY AND EXPERIMENT

689

CIRCUIT
UNDER
TEST

f ± lOOO'v

ATTENUATOR

PHASE

"shifter

ATTENUATOR

PHASE _
SHIFTER

P>^^f

-^^

^-nAA/^

AMPLIFIER
DETECTOR

Fig. 5 — Schematic diagram of the null method used to measure the transfer factor.

connected across the input is given by

PP.cos (27r-1000/).

If now these two waves are to be made to cancel, there must be a dif-
ference in the attenuation of the two branches equal to the ratio
Po/Pi, and a difference in the phase shift equal to T 0. The change in
sign of the phase angle introduced by setting the beat oscillator above
or below the test frequency is most conveniently handled by setting
the carrier oscillator consistently on the same side of the test frequency
in making a run over the frequency range.

By using a high gain amplifier preceding the detector, the precision
may be made great, limited only by circuit noise and by interference.
The attenuators and phase shifters are calibrated separately. It should
be noted that any difference in the transfer constants of the two de-
modulator circuits may be compensated by an initial adjustment which
is carried out by paralleling the input terminals of the two demodula-
tors across a source of electromotive force. With the particular type
of phase shifter used the phase shift may be changed without altering
the attenuation, so that the two settings for amplitude and phase may
be made independently.

690

BELL SYSTEM TECHNICAL JOURNAL

Visual Method

In the visual method of observation, a steady potential proportional
to the inphase component of the transfer factor is impressed across
one pair of plates of a cathode ray oscillograph and another steady
potential proportional to the quadrature component is impressed
across the other pair of plates, the constant of proportionality being
the same for the two components. In this way the transfer factor
at any frequency appears as a single point, the vector from the origin
to the displaced beam constituting the transfer factor. The locus
of all these points, i.e., vector tips, over the frequency range constitutes
the transfer factor polar diagram.

To provide rectified potentials proportional to inphase and to quad-
rature components respectively, use is made of the properties of the so-
called vacuum tube wattmeter.^" As used in practice, this device con-
sists of two triodes in push-pull connection (Fig. 6), the series arm of
the grid circuit being connected to the unknown potential, and the

Fig. 6 — Circuit of a vacuum tube wattmeter used to provide a rectified potential
proportional to the product of the two impressed grid potentials (both of the same
frequency) multiplied by the cosine of the phase angle between them.

shunt arm of the grid circuit being connected to a source of the same
frequency but of standard phase. Under these conditions the rectified
output in the plate circuit flowing in series with the two plates is pro-
portional to the product of the two impressed voltages multipled by
the cosine of the angle between them.

As shown in Fig. 7, two separate wattmeters are employed, one for
each phase, their series input terminals being connected together across
the output of the circuit under test. To the common branch of one of
these wattmeters is supplied the same potential as is fed to the input
of the circuit under test. The rectified output of this wattmeter there-
fore is proportional to the product of the input and output voltages
multiplied by the cosine of the transfer factor phase angle. This po-

10 U. S. Patent 1,586,533; Turner and McNamara, Proc. I. R. E., vol. 18, p. 1743;
October (1930).

REGENERATION THEORY AND EXPERIMENT

691

tential is supplied to those plates of the oscillograph which produce a
horizontal deflection. To the common branch of the other wattmeter
is applied a potential equal in amplitude to the input voltage but lag-
ging behind it by 90 degrees. The rectified output of this wattmeter is
proportional to the product of input and output voltages multiplied
by the cosine of the transfer factor phase angle minus 90 degrees, or in
other words proportional to the sine of the transfer factor phase angle.
This voltage is supplied to those plates of the oscillograph which pro-
duce a vertical deflection. We have then across one pair of plates of
the oscillograph a steady potential proportional to the real component

FROM

OSCILLATOR

PHASE t

CIRCUIT
UNDER
TEST

-\i\r

WATT
METER

CATHODE RAY ,

OSCILLOGRAPH

/ 1 \

hjf

WATT
METER

OSCILLATOR
PHASE 2

Fig. 7 — Schematic diagram of the circuit used to plot the transfer factor diagram on
the screen of a cathode ray oscillograph.

of the transfer factor, and across the other pair of plates we have im-
pressed a steady potential proportional to the imaginary component
of the transfer factor. These two components act upon the beam of the
oscillograph to produce a deflection which in amplitude and in phase is
the resultant of the two component deflections and so corresponds to
the transfer factor.

It will be observed that the above procedure requires a two-phase
source of constant amplitude, the frequency of which is variable over
the range necessary to establish the properties of the amplifier. In the
present instance the frequency range extends from 0.5 to 30 kilocycles,
and the accuracy required is of the order of five per cent.

A schematic of the two-phase oscillator used is shown in Fig. 8.
This oscillator is of the heterodyne type. Two independent sources
are used, one of constant frequency (100 kilocycles), the other variable
in frequency and practically constant in amplitude over the range of
100 to 130 kilocycles. As indicated in the figure, the variable fre-

692

BELL SYSTEM TECHNICAL JOURNAL

^Qib-vQil^'

CW^^W\

100-130 KC
OSCILLATOR

LOW PASS
FILTER

AMPLIFIER

6 6

PHASE 1

0.5-30 KC

PHASE 2
0.5-30 KC

9 9

LOW PASS
FILTER

AMPLIFIER

Fig. 8 — Circuit diagram of a heterodyne type two-phase oscillator, the output
frequency of which is continuously variable from 0.5 to 30 kilocycles. The output
of each phase and the 90-degree difference between the two phases are practically
constant over the frequency range.

quency oscillator is connected to the common branches of the two push-
pull modulators. The fixed frequency oscillator is connected in series
with the grid circuits of the two modulators. The resistance-capacity
networks shown in the circuits of the fixed frequency oscillator are pro-
vided to produce phase shifts of 90 degrees between the two series
voltages of the two modulators. In the same manner as that discussed
before in connection with the null method measuring circuit, the phase
shift introduced to the fixed frequency is maintained in the beat fre-
quency output, so that the phase difference of 90 degrees is preserved
in the outputs of the two modulators when the variable frequency
oscillator goes from about 100.5 to 130 kilocycles. The outputs of the
two phases are connected to the test amplifier and to the wattmeters as
shown in the preceding Fig. 7.

Comparison of the Methods
Measurements of transfer factors by the two methods outlined
above were found to be in agreement within the error of measurement.
The visual method as developed was capable of use over only a very re-

REGENERATION THEORY AND EXPERIMENT

693

stricted frequency range as compared to the null method, but it
covered the region of particular interest in the experiments conducted
for the purpose of testing the stability criterion. Through its use,
measurements over its frequency range could be made in a few minutes
time, whereas corresponding measurements by the more precise null
method required three to six hours. Of course the time intervals cited
do not include time occupied in setting up and adjusting the apparatus.

Test Amplifier and Experimental Results
Test Amplifier

The stability criterion indicates three distinct conditions of in-
terest, one of which is unstable, the other two being stable. The un-
stable condition (1) is that in which the transfer factor curve encloses
the point (1, 0). Two stable conditions are those in which (1, 0) is not
enclosed by the curve, but in which (2) the curve crosses the zero phase
shift axis at points greater than unity, and (3) the curve does not cross
the zero phase shift axis at points greater than unity. Condition (2)
is of particular interest because while it is judged stable on the basis
of Nyquist's criterion it would appear to be unstable on the basis of the
older transfer criterion discussed in the first and second sections.

For test purposes an amplifier was designed which, upon variation
of an attenuator in the feed-back path, would satisfy each of the three
above conditions in turn. The amplifier schematic is shown in Fig. 9.

Fig. 9 — Circuit diagram of the feed-back amplifier used in testing the stability
criterion. The dashed line indicates the point at which the loop was broken for
measurement of the transfer factor. At the left of this line is shown the resistance
attenuator provided to vary the gain around the feed-back loop.

694 BELL SYSTEM TECHNICAL JOURNAL

It has three stages, the first two tubes being space charge grid pentodes,
and the last one a triode. The interstage coupHng circuits were made
up of simple inductances and resistances as shown. The amplifier was
designed by E. L. Norton and E. E. Aldrich to provide a transfer factor
characteristic having the desired shape, i.e., a loop crossing the zero
phase axis in the neighborhood of 10 kilocycles. It will be observed
that the feed-back circuit is connected between bridge networks in
both input and output circuits, which were provided to eliminate reac-
tion of the input and output circuits upon the feed-back network.^^

Experimental Results

The transfer factor was measured for a zero setting of the feed-back
attenuator over a frequency range of 0.5 to 1200 kilocycles. The re-
sults are shown in Fig. 10. The method of plotting this figure requires
some discussion. In order to keep the curve within a reasonable size
and still show the necessary details the scale has been made logarithmic
by plotting the gain around the loop in decibels instead of the corre-
sponding numerical ratios. It is of course impossible to carry this out
completely on a polar diagram since the transfer factor goes to zero at
high frequencies. To take care of this the scale is made logarithmic
only above zero gain, corresponding to unit transfer ratio, and is linear
below. It should be noted that if the logarithmic portion of the scale
is translated outward so that the zero decibel point lies successively in
the regions marked A, B, C, and D, the indicated amplifier conditions
correspond to those designated above as (1), (2), (1) and (3) respec-
tively. Experimentally an increase of the feed-back attenuator
corresponds to such a translation of the logarithmic scale by an amount
equal to the increase in attenuation. Therefore, the transition from
one condition to another should occur when the attenuator setting is
equal to the gain at a zero phase point in the curve as measured with a
zero attenuator setting.

The test of the stability criterion consists of a determination of the
attenuator settings at which oscillations begin, and a comparison of
these settings with those at which a transition from a stable to an un-
stable condition is predicted by the theory. Experimentally oscilla-
tions were found to occur in regions A and C and not in regions B and D
which is in qualitative agreement with Nyquist's predictions. Quanti-
tatively the measured and predicted transition points agreed within
one decibel which is estimated to be within the experimental error.

It should be noted that the plotted curve has been drawn up for
A(jo}), no points of ^(— jco) being shown, although both are required

1' H. S. Black, Bell Sys. Tech. Jour., January, 1934.

REGENERATION THEORY AND EXPERIMENT

695

11 KcT---^,,^ / /\

/TtC / /\\'° ^^

vA\

""t$

/4i<^oo _-— \— ^

/ 'LOOP GAIN IN DB

" 7

- ^^v

(U^^^^C

80 60 40

-—^2^

/ /

-TO 00

240°

?0° 300°

0°
360°

Fig. 10 — Transfer factor diagram for the amplifier of Fig. 9 with the feed-back
attenuator set at zero decibel.

by the theoretical derivation. Where the transfer factor is zero at zero
frequency, only A{jiS) is required since the loop then closes for positive
values of w. In amplifiers transmitting d.c. , however, both positive and
negative values of oj are needed ^ to form a closed loop. In any case
■4(— jco) is the mirror image of ^(jco) about the x-axis.

Extensions of the Criterion
Nonlinear Amplifier
The stability criterion which was verified by the experiments re-
ported in the preceding section is framed for linear systems, those in
which the steady state response is linearly proportional to the applied
force. In vacuum tube circuits, linearity is best approximated at small
force amplitudes, and is departed from to an extent dependent upon
the impressed potentials, as well as upon tube and circuit characteris-
tics. The divergence from linearity becomes well marked when the
load capacities of the tubes are approached, or when grid current is
made to flow through large grid impedances. The question then arises
as to the form which the stability criterion takes when a tube circuit is

* Log. cit.

696 BELL SYSTEM TECHNICAL JOURNAL

operated in a nonlinear region — let us say by impressing upon the cir-
cuit a sufficiently large alternating potential provided by an external
independent generator.

To answer this question we may consider the response of the am-
plifier, loaded by the independent generator, to a small alternating
potential introduced for test purposes. Since the response of the sys-
tem is known to be linear from the theory of perturbations, we might
attempt to apply the linear criterion to the small superposed force. To
do this it is necessary to measure the transfer factor for the small super-
posed force over the frequency range at a particular load of interest.

Application of the experimental technique to this extended criterion
introduces difficulties since the opening of the feed-back loop for meas-
uring purposes disturbs to a certain extent the distribution of these
loads, particularly the harmonics, and modulation products in general.
This makes it difficult to get the same loading effect when the loop is
opened for measuring purposes as obtained when the loop is closed.
Another consideration is that the response to the small component
may be expected to vary in general at different points on the loading
wave, so that the measuring procedure averages the response over a
cycle of the loading wave. A method of measurement analogous to
that of the flutter bridge would be required to evaluate the transfer
factor at points of the loading cycle. Further, the measuring appara-
tus is affected by the presence of the loading currents when these are
sufficiently large. In the present case in which the loading frequency
(60 kilocycles) was far removed in the frequency scale from the test
frequencies, it was found possible to approximate the necessary meas-
urements by the insertion of selective circuits.

The curves of Fig. 1 1 represent portions of the transfer factor polar
diagram for an amplifier similar to the one previously described, meas-
ured by the visual method with different loading amplitudes. The
effect of the load on this particular amplifier is to change both phase
shift and amplitude so that the curves shrink both radially and tan-
gentially, pulling the loop back across the zero phase axis until, at the
heaviest load, the two low-frequency crossings are completely elimin-
ated. If the extended criterion is valid, we should expect the ampli-
fier to be stable at any setting of the feed-back attenuator. As the
load is decreased from this value, the crossings occur at successively
higher gains so that the start of oscillations would occur at progres-
sively higher settings of the feed-back attenuator.

The curves of Fig. 12 show the attenuator settings predicted by the
extended criterion and those determined by direct observation of the
attenuator setting required for oscillations when the feed-back circuit

REGENERATION THEORY AND EXPERIMENT

697

120°

90°

60°

< vsv

-V-

•-

"nj

/\ •'v/nT^n

/ r\/ JK

150°

Sli

/ / / /\ / "^ ^

// / Af \^x^ W--^

LOOP GAIN

^^-f— -0>

^^^-^c"^

180°

\ T

~ T^A

Y^^^^^^^^tT' ; ;

IN DB

L — t

Z-\<J

Tb— 10 130 ' / 50 ,'

__ — ■ LOAD IN WATTS ^^

^^SoJ^-f^

-__/^

0.0 -^y

/yUj

rw^NcT^-^^-A^

/ ~~" /

/jT

\ \ \ >( ^~^ -A^""^

r-^ /

/^/

^ / A

210°

\ /n/ /

I-

—

30°

0
360°

240 270 300

Fig. 11 — The transfer factor diagram for the amplifier of Fig. 9 with the feed-back
attenuator set at zero decibel. The four curves shown correspond to different
amounts of the 60-kilocycle load.

was closed. Two sets of curves are shown, one for each of the low-
frequency crossings. These are plotted against the loading amplitude.
The agreement between the experimental and predicted values is close
for the higher gain crossing at small loading amplitudes, but a di-
vergence is apparent at high loads. For the lower crossing there is a
divergence of 1.5 decibels at low loads, which changes sign and be-
comes greater at the higher loads. These divergences may be ascribed
to a variety of causes among which probably the most important are
the effects of harmonics upon the amplifier loading, overloading of the
measuring apparatus by harmonics of the loading electromotive force,
and phase shifts introduced by the selective circuits. The last two
causes may be eliminated by improved technique, but the first cause in
general introduces a fundamental difficulty, particularly important
when large nonlinearities are involved.

Negative Impedances

One of the early forms of stability criterion mentioned in the first
section was that relating to the measured impedance of the circuit.

698

BELL SYSTEM TECHNICAL JOURNAL

70
65

~^^

, ^

NG POINTS
SING POINTS

60
56
50

CR03

45
40
35

— —

_ ^^

~"~-N

GRID OF LAST TUBE
GOES POSITIVE ~1

0 5 10 15 20 25 30

LOAD CURRENT-DB UP ON I MIL INTO 600 OHMS

Fig. 12 — Comparison of feed-back attenuator settings required for the starting
of oscillations, and those deduced from the transfer diagram, plotted as functions of
the 60-kilocycle load. The two dashed curves correspond to the two points (roughly
4.5 and 8 kilocycles) at which the transfer factor diagram (Fig. 11) crosses the zero
phase axis. The gains of Figs. 11 and 12 cannot be compared directly because of a
change made in the amplifier circuit of Fig. 12 which increased the loop gain.

Nyquist's criterion involving the transfer factor may be transformed so
as to formulate a more complete criterion involving such an impedance.
To do this we have to express the factor (1 —A), on which the
stability criterion was based, in terms of the circuit impedances. For
illustrative purposes we may quote the results obtained with the two
fundamental forms of feed-back circuits, the series and shunt types. ^^
These results, while obtained for the input circuit of the amplifier, are
valid for any other point of the feed-back loop. Further, combinations
of the shunt and series type feed-back circuits may be used.

Series Feed-Back

The series circuit is shown in Fig. 13, so called because the feed-
back is applied in series with the amplied electromotive force and the
amplifier input. The passive impedances marked are those existing
when the feed-back loop is broken and terminated as indicated by the

^2 Crisson, Bell Sys. Tech. Jour., vol. X, p. 485.

REGENERATION THEORY AND EXPERIMENT

699

dotted lines. By direct circuit analysis, the current and voltage ampli-
tudes in the feed-back condition are related by

E= (Z + Zo + Z,)(l - A)r,

where A and the Z's are functions of frequency. The total effective
circuit impedance is obtained as the multiplifier of / in the right mem-

AMPLI-
FIER

2|N — *- 1
I 1

^v-*-

\V V

>• z

O/ ) E

NET-
WORK

zo —

'v V V

^^ \

Fig. 13 — Series type feed-back circuit. The dotted resistances indicate the termi-
nations applied when the feed-back circuit was broken, to which the passive impe-
dances {ZiZo) apply. Zi„ represents the effective input impedance with the feed-back
circuit connected through.

ber. Subtracting the generator impedance Z from the total, the input
impedance becomes

Z/^ = (Zo-f Zi)(l - A) ~ AZ,

from which.

A =

Z + Zo + z,

1 +

Zi.

Of the two factors of the right member, the first one, involving pas-
sive impedances alone, can have no roots with positive real part. Any
such roots must, therefore, be contained in the second bracketed fac-
tor and then only when Ztn is negative. Hence paraphrasing the
transfer factor criterion, if we plot — Zin/Z over the frequency range,
the circuit is stable when the point (1, 0) is not enclosed by the result-
ant curve.

Shunt Feed-Back
Proceeding as in the series case with the circuit of Fig. 14 we get

AMPLI-
FIER

Zl —

S z

Z|N — *-|

I 1

V V ^

Oj ) E

NET-
WORK

0-^

V V \r"

Fig. 14 — Shunt type feed-back circuit. The notation corresponds to that of Fig. 13.

700 BELL SYSTEM TECHNICAL JOURNAL

\ — A = ^° I 1 + —

where Za represents the impedance of Zq and Zi in parallel. Again
only the bracketed term can yield undamped transients so that the
criterion involves plotting — ZjZin over the frequency range; if the
resultant curve does not enclose (1,0) the circuit is stable.

It may be remarked that these results are applicable to circuits in-
cluding two-terminal negative impedances such as the oscillating arc
and the dynatron, which are of the series and the shunt type respec-
tively.

Acknowledgments

The authors are indebted to Mr. L. W. Hussey for discussions of
theoretical points, and to Mr. P. A. Reiling for his cooperation in the
experiments.

Abstracts of Technical Articles from Bell System Sources

Shared Channel Broadcasting} C. B. Aiken. This paper deals
with the experimental studies made on the character and causes of
interference noticeable in shared channel broadcasting, such as hetero-
dyning, flutter, sideband interference and wobbling. Valuable data are
included on the characteristics of square-law and linear detectors anent
to interference.

The Determination of Dielectric Properties at Very High Frequencies.^
J. G. Chaffee. A simple method of determining the dielectric con-
stant and power factor of solid dielectrics at frequencies as high as 20
megacycles, with an accuracy which is sufficient for most purposes,
is described. The major sources of error are discussed in detail, and
several precautions which should be observed are pointed out.

Measurements of the dielectric properties at 18 megacycles of a
number of commonly used materials have shown that in general the
power factor and dielectric constant are not widely different from those
which obtain at frequencies of the order of one megacycle.

In addition, the results of an investigation of the input impedance of
vacuum tube voltmeters at high frequencies are described as an illus-
tration of the further application of this method of measurement.

Optical Factors in Caesium- Silver-Oxide Photoelectric Cells.^ H. E.
Ives and A. R. Olpin. This paper describes an investigation of the
part played by the angle of incidence and state of polarization of the
exciting light in producing the enhanced or selective emission of photo-
electrons in the red region of the spectrum which is characteristic of
photoelectric cells made by treating a silver surface with oxygen and
caesium vapor (Fig. 1). This question is one which has been raised in
connection with all types of photoelectric cells having composite sur-
faces and which exhibit spectrally selective emission. It has thus been
an open question whether the selective peaks in the spectral response
curves exhibited by the alkali hydride cells are to be ascribed to an
enhanced effect of the perpendicular vector of obliquely incident

^ Radio Engineering, June, 1934.

2 Proc. I. R. E., August, 1934.

* Jour. Op. Soc. Am., August, 1934.

701

702 BELL SYSTEM TECHNICAL JOURNAL

radiation, or whether the spectral selectivity is in the nature of a locally
intrinsic emissive power, such as would be caused by an optical absorp-
tion band or an electronic transmission band. In order to answer this
question, it is necessary to have emitting surfaces of a specular char-
acter. Such surfaces have not been prepared with the alkali hydrides,
but it has been found possible to make the caesium-silver-oxide cells
on specular plates of silver so that they retain their specular character
in the final sensitized surface. Cells of this sort were used in this study,
and have made possible a clear separation of the emissive singularities
due to optical conditions and the singularities which may be described
as intrinsic to the material.

Both from their method of preparation and from their optical be-
havior, we have felt justified in considering the caesium-silver-oxide
photoelectric cells prepared with specular silver surfaces as consisting
of silver surfaces overlaid with a thick layer of transparent refracting
material, on the top of which is a thin photosensitive layer. The
silver plates, after oxidation, exhibit interference colors, the exact
color depending upon the amount of oxidation. Viewed at an angle
through a nicol prism, these oxidized plates exhibit the well-known
properties of thin refractive layers on a metal base. Thus when the
plane of polarization is changed from the plane of incidence to the
plane perpendicular thereto, no change of hue takes place for small
angles of incidence; but at large angles, the color changes to a comple-
mentary hue. After the silver oxide surface has been exposed to cae-
sium vapor and given a heat treatment, these optical properties are
still usually observable, but degraded. The softening of the interfer-
ence colors may be due either to a change in thickness of the refracting
medium as caesium oxide is formed or to the introduction of a general
body color. In a few less common cases the colors faded out com-
pletely, the plate at the end of the heat treatment being metallic in
appearance yet still exhibiting a pronounced selective response to red
and infrared light.

The behavior of a thin photoelectric sheet separated from a specular
metal surface by a layer of refracting medium has been treated in an
earlier paper where a layer of caesium was deposited on the top of a
quartz-coated platinum plate. The data obtained in this earlier paper
are immediately applicable to the present problem, granting the similar-
ity of conditions which we have assumed. It has been convenient to
pursue this present study on the assumption of such a similarity and to
arrive at conclusions from the agreement with, or deviation from, the
results obtained from the simpler materials and conditions previously
studied.

ABSTRACTS OF TECHNICAL ARTICLES 703

Phase Angle of Vacuum Tube Tramconductance at Very High Fre-
quencies.* F. B. Llewellyn. Theoretical considerations indicate
that the transconductance of a vacuum tube exhibits a phase angle
when the transit time of electrons from cathode to anode becomes an
appreciable fraction of the high-frequency period. Measurements
show that such a phase angle actually occurs and that its behavior is in
general agreement with the theoretical predictions.

Application of Sound Measuring Instruments to the Study of Phonetic
Problems.^ John C. Steinberg. This paper gives the results of a
period by period analysis of the vowel sound waves occurring when the
sentence "Joe took father's shoe bench out" was spoken. Such an
analysis gives an approximate picture of the time variations in r.m.s.
amplitude of the wave, frequency of voice fundamental, and frequency
regions of overtone reenforcement. Although the study is confined to
a few sounds and one speaker's voice, it illustrates a method of ap-
proach to studies of speech production and measurement.

*Proc. I. R. E., August, 1934.

^ Jour. Acous. Soc. Am., July, 1934.

Contributors to this Issue

Lloyd Espenschied. Mr. Espenschied is High Frequency Trans-
mission Development Director in the Bell Telephone Laboratories.
He joined the Bell System in 1910, having graduated from Pratt
Institute the previous year. He has taken an important part in prac-
tically all of the Bell System radio developments, beginning with the
first long-distance radio-telephone tests of 1915, at which time he re-
ceived the voice in Hawaii from Arlington, Virginia. He has partici-
pated in a number of international conferences on electric communica-
tions.

J. G. Kreer, B.S. in Electrical Engineering, University of Illinois,
1925 ; M.A., Columbia University, 1928. Bell Telephone Laboratories,
1925-. Mr. Kreer has been engaged in research work on carrier fre-
quency systems.

S. A. Levin, E.E., Chalmers Technical Institute, Gothenburg, 1919;
Technische Hochschule, Berlin, 1920-21; Technische Hochschule,
Dresden, 1921-23. Radio Department, General Electric Company,
Schenectady, N. Y., 1923-26; Engineering Department, National
Electric Light Association, New York, N. Y., 1926-30. Bell Telephone
Laboratories, 1930-. Mr. Levin's work has to do with the develop-
ment of high-frequency measuring equipment for carrier systems.

G. L. Pearson, A.B., Willamette University, 1926; M.A. Stanford
University, 1929. Bell Telephone Laboratories, 1929-. Mr. Pearson
has been engaged in a study of the noise inherent in electric circuits.

D. B. Penick, B.S. in Electrical Engineering, University of Texas,
1923; B.A., 1924; M.A. in Physics, Columbia University, 1927. West-
ern Electric Company, Engineering Department, 1924-25; Bell Tele-
phone Laboratories, 1925-. Mr. Penick has been engaged in special
problems related to the development of vacuum tubes.

704

CONTRIBUTORS TO THIS ISSUE 705

E. Peterson, Cornell University, 1911-14; Brooklyn Polytechnic,
E.E., 1917; Columbia, A.M., 1923; Ph.D., 1926; Electrical Testing
Laboratories, 1915-17; Signal Corps, U. S. Army, 1917-19. Bell
Telephone Laboratories, 1919-. Dr. Peterson's work has been largely
in theoretical studies of carrier current apparatus.

Liss C. Peterson, E.E., Chalmers Technical Institute, Gothenburg,
1920; Technische Hochschule, Charlottenburg, 1920-21; Technische
Hochschule, Dresden, 1921-22; Signal Corps, Swedish Army, 1922-23.
American Telephone and Telegraph Company, 1925-30 ; Bell Telephone
Laboratories, 1930-. Mr. Peterson is engaged in the study of modula-
tion and other problems connected with high frequency carrier systems.

S. A. ScHELKUNOFF, B.A., M.A., in Mathematics, The State College
of Washington, 1923; Ph.D. in Mathematics, Columbia University,
1928. Engineering Department, Western Electric Company, 1923-25.
Bell Telephone Laboratories, 1925-26. Department of Mathematics,
State College of Washington, 1926-29. Bell Telephone Laboratories,
1929-. Dr. Schelkunoff has been engaged in mathematical research,
especially in the field of electromagnetic theory.

M. E. Strieby, A.B., Colorado College, 1914; B.S., Harvard, 1916;
B.S. in E.E., M.LT., 1916; New York Telephone Company, Engineer-
ing Department, 1916-17; Captain, Signal Corps, U. S. Army,
A. E. P., 1917-19. American Telephone and Telegraph Company,
Department of Development and Research, 1919-29; Bell Telephone
Laboratories, 1929-. Mr. Strieby has been associated with various
phases of transmission work, more particularly with the development
of long toll circuits. At the present time, in his capacity as Carrier
Transmission Research Engineer, he directs studies of new and im-
proved methods of carrier frequency transmission over existing or new
facilities.

L. A. Ware, B.E., Engineering College, University of Iowa, 1926;
M.S., University of Iowa, 1927; PhD., Physics Department, University
of low^a, 1930. Instructor in Physics, University of Iowa, 1926-29.
Bell Telephone Laboratories, 1929-. Dr. Ware's work has been chiefly
in connection with regenerative amplifier development.

.\**.^^v

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